Difference between revisions of "Linear Independence of Functions"
(Created page with "<p>If we have an <math>n^{\mathrm{th}}</math> order linear homogenous differential equation <math>\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac...") |
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<p>If we have an <math>n^{\mathrm{th}}</math> order linear homogenous differential equation <math>\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0</math> where <math>p_1</math>, <math>p_2</math>, …, <math>p_n</math> are continuous on an open interval <math>I</math> and if <math>y = y_1(t)</math>, <math>y = y_2(t)</math>, …, <math>y = y_n(t)</math> are solutions to this differential equation, then provided that <math>W(y_1, y_2, ..., y_n) \neq 0</math> for at least one point <math>t \in I</math>, then <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> form a fundamental set of solutions to this differential equation - that is, for constants <math>C_1</math>, <math>C_2</math>, …, <math>C_n</math>, then every solution to this differential equation can be written in the form:</p> | <p>If we have an <math>n^{\mathrm{th}}</math> order linear homogenous differential equation <math>\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0</math> where <math>p_1</math>, <math>p_2</math>, …, <math>p_n</math> are continuous on an open interval <math>I</math> and if <math>y = y_1(t)</math>, <math>y = y_2(t)</math>, …, <math>y = y_n(t)</math> are solutions to this differential equation, then provided that <math>W(y_1, y_2, ..., y_n) \neq 0</math> for at least one point <math>t \in I</math>, then <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> form a fundamental set of solutions to this differential equation - that is, for constants <math>C_1</math>, <math>C_2</math>, …, <math>C_n</math>, then every solution to this differential equation can be written in the form:</p> | ||
| − | < | + | <math>\begin{align} \quad y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t) \end{align}</math> |
<p>We will now look at the connection between the solutions <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> forming a fundamental set of solutions and the linear independence/dependence of such solutions. We first define linear independence and linear dependence below.</p> | <p>We will now look at the connection between the solutions <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> forming a fundamental set of solutions and the linear independence/dependence of such solutions. We first define linear independence and linear dependence below.</p> | ||
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<td><strong>Definition:</strong> The functions <math>f_1</math>, <math>f_2</math>, …, <math>f_n</math> are said to be <strong>Linearly Independent</strong> on an interval <math>I</math> if for constants <math>k_1</math>, <math>k_2</math>, …, <math>k_n</math> we have that <math>k_1f_1(t) + k_2f_2(t) + ... + k_nf_n(t) = 0</math> implies that <math>k_1 = k_2 = ... = k_n = 0</math> for all <math>t \in I</math>. This set of functions is said to be <strong>Linearly Dependent</strong> if <math>k_1f_1(t) + k_2f_2(t) + ... + k_nf_n(t) = 0</math> where <math>k_1</math>, <math>k_2</math>, …, <math>k_n</math> are not all zero for all <math>t \in I</math>.</td> | <td><strong>Definition:</strong> The functions <math>f_1</math>, <math>f_2</math>, …, <math>f_n</math> are said to be <strong>Linearly Independent</strong> on an interval <math>I</math> if for constants <math>k_1</math>, <math>k_2</math>, …, <math>k_n</math> we have that <math>k_1f_1(t) + k_2f_2(t) + ... + k_nf_n(t) = 0</math> implies that <math>k_1 = k_2 = ... = k_n = 0</math> for all <math>t \in I</math>. This set of functions is said to be <strong>Linearly Dependent</strong> if <math>k_1f_1(t) + k_2f_2(t) + ... + k_nf_n(t) = 0</math> where <math>k_1</math>, <math>k_2</math>, …, <math>k_n</math> are not all zero for all <math>t \in I</math>.</td> | ||
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<p>Perhaps the simplest linearly independent sets of functions is that set that contains <math>f_1(t) = 1</math>, <math>f_2(t) = t</math>, and <math>f_3(t) = t^2</math>. Let <math>k_1</math>, <math>k_2</math>, and <math>k_3</math> be constants and consider the following equation:</p> | <p>Perhaps the simplest linearly independent sets of functions is that set that contains <math>f_1(t) = 1</math>, <math>f_2(t) = t</math>, and <math>f_3(t) = t^2</math>. Let <math>k_1</math>, <math>k_2</math>, and <math>k_3</math> be constants and consider the following equation:</p> | ||
| − | < | + | <math>\begin{align} \quad k_1f_1(t) + k_2f_2(t) + k_3f_3(t) = 0 \\ \quad k_1 + k_2t + k_3t^2 = 0 \end{align}</math> |
<p>It's not hard to see that equation above is satisfied if and only if the constants <math>k_1 = k_2 = k_3 = 0</math>.</p> | <p>It's not hard to see that equation above is satisfied if and only if the constants <math>k_1 = k_2 = k_3 = 0</math>.</p> | ||
<p>For another example, consider the functions <math>f_1(t) = \sin t</math> and <math>f_2(t) = \sin (t + \pi)</math> defined on all of <math>\mathbb{R}</math>. This set of functions is not linearly independent. To show this, let <math>k_1</math> and <math>k_2</math> be constants and consider the following equation:</p> | <p>For another example, consider the functions <math>f_1(t) = \sin t</math> and <math>f_2(t) = \sin (t + \pi)</math> defined on all of <math>\mathbb{R}</math>. This set of functions is not linearly independent. To show this, let <math>k_1</math> and <math>k_2</math> be constants and consider the following equation:</p> | ||
| − | < | + | <math>\begin{align} \quad k_1f_1(t) + k_2f_2(t) = 0 \\ \quad k_1 \sin t + k_2 \sin (t + \pi) = 0 \end{align}</math> |
<p>Now choose <math>t = \pi</math>. Then we have that:</p> | <p>Now choose <math>t = \pi</math>. Then we have that:</p> | ||
| − | < | + | <math>\begin{align} \quad k_1 \sin \pi + k_2 \sin 2\pi = 0 \\ \end{align}</math> |
<p>But the above equation is true for any choice of constants <math>k_1</math> and <math>k_2</math> since <math>\sin \pi = \sin 2\pi = 0</math>, and thus <math>f_1</math> and <math>f_2</math> do not form a linearly independent set on all of <math>\mathbb{R}</math>.</p> | <p>But the above equation is true for any choice of constants <math>k_1</math> and <math>k_2</math> since <math>\sin \pi = \sin 2\pi = 0</math>, and thus <math>f_1</math> and <math>f_2</math> do not form a linearly independent set on all of <math>\mathbb{R}</math>.</p> | ||
<p>From the concept of linear independence/dependence, we obtain the following theorem on fundamental sets of solutions for <math>n^{\mathrm{th}}</math> order linear homogenous differential equations.</p> | <p>From the concept of linear independence/dependence, we obtain the following theorem on fundamental sets of solutions for <math>n^{\mathrm{th}}</math> order linear homogenous differential equations.</p> | ||
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<td><strong>Theorem 1:</strong> Let <math>\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0</math> be an <math>n^{\mathrm{th}}</math> order linear homogenous differential equation. If <math>y = y_1(t)</math>, <math>y = y_2(t)</math>, …, <math>y = y_n(t)</math> are solutions to this differential equation then <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> form a fundamental set of solutions to this differential equation on the open interval <math>I</math> if and only if <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly dependent on <math>I</math>.</td> | <td><strong>Theorem 1:</strong> Let <math>\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0</math> be an <math>n^{\mathrm{th}}</math> order linear homogenous differential equation. If <math>y = y_1(t)</math>, <math>y = y_2(t)</math>, …, <math>y = y_n(t)</math> are solutions to this differential equation then <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> form a fundamental set of solutions to this differential equation on the open interval <math>I</math> if and only if <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly dependent on <math>I</math>.</td> | ||
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<ul> | <ul> | ||
<li><strong>Proof:</strong> Consider the following <math>n^{\mathrm{th}}</math> order linear homogenous differential equation:</li> | <li><strong>Proof:</strong> Consider the following <math>n^{\mathrm{th}}</math> order linear homogenous differential equation:</li> | ||
</ul> | </ul> | ||
| − | < | + | <math>\begin{align} \quad \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0 \end{align}</math> |
<ul> | <ul> | ||
<li><math>\Rightarrow</math> Suppose that <math>y = y_1(t)</math>, <math>y = y_2(t)</math>, …, <math>y = y_n(t)</math> form a fundamental set of solutions to this differential equation on the open interval <math>I</math>. Then this implies that for all <math>t_0 \in I</math> we have that :</li> | <li><math>\Rightarrow</math> Suppose that <math>y = y_1(t)</math>, <math>y = y_2(t)</math>, …, <math>y = y_n(t)</math> form a fundamental set of solutions to this differential equation on the open interval <math>I</math>. Then this implies that for all <math>t_0 \in I</math> we have that :</li> | ||
</ul> | </ul> | ||
| − | < | + | <math>\begin{align} \quad W(y_1, y_2, ..., y_n) \right|_{t_0} \neq 0 \end{align}</math> |
<ul> | <ul> | ||
<li>Thus this implies that following system of equations have only trivial solution <math>k_1 = k_2 = ... = k_n = 0</math>:</li> | <li>Thus this implies that following system of equations have only trivial solution <math>k_1 = k_2 = ... = k_n = 0</math>:</li> | ||
</ul> | </ul> | ||
| − | < | + | <math>\begin{align} \quad k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0 \\ \quad k_1y_1'(t) + k_2y_2'(t) + ... + k_ny_n'(t) = 0 \\ \quad \quad \quad \quad \quad \quad \vdots \quad \quad \quad \quad \quad \quad \\ \quad k_1y_1^{(n-2)}(t) + k_2y_2^{(n-2)}(t) + ... + k_ny_n^{(n-2)}(t) = 0 \\ \quad k_1y_1^{(n-1)}(t) + k_2y_2^{(n-1)}(t) + ... + k_ny_n^{(n-1)}(t) = 0 \end{align}</math> |
<ul> | <ul> | ||
<li>Thus the equation <math>k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0</math> implies that <math>k_1 = k_2 = ... = k_n = 0</math>. Thus <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly independent on <math>I</math>.</li> | <li>Thus the equation <math>k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0</math> implies that <math>k_1 = k_2 = ... = k_n = 0</math>. Thus <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly independent on <math>I</math>.</li> | ||
</ul> | </ul> | ||
<ul> | <ul> | ||
| − | <li><math>\Leftarrow</math> We will prove the converse of Theorem 1 by contradiction. Suppose that <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly independent on <math>I</math>, and assume that instead <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> do NOT form a fundamental set of solutions on <math>I</math>. Then for some <math>t_0 \in I</math>, the Wronskian <math>W(y_1, y_2, ..., y_n) \ | + | <li><math>\Leftarrow</math> We will prove the converse of Theorem 1 by contradiction. Suppose that <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly independent on <math>I</math>, and assume that instead <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> do NOT form a fundamental set of solutions on <math>I</math>. Then for some <math>t_0 \in I</math>, the Wronskian <math>W(y_1, y_2, ..., y_n) \right|_{t_0} = 0</math>. Thus the system of equations above does not have only the trivial solution. Let the constants <math>k_1^*</math>, <math>k_2^*</math>, …, <math>k_n^*</math> be a nontrivial solution to this system. Define <math>\phi(t)</math> as:</li> |
</ul> | </ul> | ||
| − | < | + | <math>\begin{align} \quad \phi(t) = k_1^* y_1(t) + k_2^* y_2(t) + ... + k_n^* y_n(t) \end{align}</math> |
<ul> | <ul> | ||
<li>Note that <math>y = \phi(t)</math> satisfies the initial conditions <math>y(t_0) = 0</math>, <math>y'(t_0) = 0</math>, …, <math>y^{(n-1)} (t_0) = 0</math>, and <math>\phi(t)</math> satisfies our <math>n^{\mathrm{th}}</math> order linear homogenous differential equation because <math>\phi(t)</math> is a linear combination of the solutions <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math>.</li> | <li>Note that <math>y = \phi(t)</math> satisfies the initial conditions <math>y(t_0) = 0</math>, <math>y'(t_0) = 0</math>, …, <math>y^{(n-1)} (t_0) = 0</math>, and <math>\phi(t)</math> satisfies our <math>n^{\mathrm{th}}</math> order linear homogenous differential equation because <math>\phi(t)</math> is a linear combination of the solutions <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math>.</li> | ||
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</ul> | </ul> | ||
| − | < | + | <math>\begin{align} \quad 0 = k_1^* y_1(t) + k_2^* y_2(t) + ... + k_n^* y_n(t) \end{align}</math> |
<ul> | <ul> | ||
<li>But <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly independent which implies that <math>k_1^* = k_2^* = ... = k_n^* = 0</math>. Thus <math>k_1^*</math>, <math>k_2^*</math>, …, <math>k_n^*</math> is a trivial solution to the system above, which is a contradiction. Therefore our assumption that <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> do not form a fundamental set of solutions was false. <math>\blacksquare</math></li> | <li>But <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> are linearly independent which implies that <math>k_1^* = k_2^* = ... = k_n^* = 0</math>. Thus <math>k_1^*</math>, <math>k_2^*</math>, …, <math>k_n^*</math> is a trivial solution to the system above, which is a contradiction. Therefore our assumption that <math>y_1</math>, <math>y_2</math>, …, <math>y_n</math> do not form a fundamental set of solutions was false. <math>\blacksquare</math></li> | ||
</ul> | </ul> | ||
Revision as of 17:55, 28 October 2021
If we have an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0} where , , …, are continuous on an open interval and if , , …, are solutions to this differential equation, then provided that for at least one point , then , , …, form a fundamental set of solutions to this differential equation - that is, for constants , , …, , then every solution to this differential equation can be written in the form:
We will now look at the connection between the solutions , , …, forming a fundamental set of solutions and the linear independence/dependence of such solutions. We first define linear independence and linear dependence below.
Definition: The functions , , …, are said to be Linearly Independent on an interval if for constants , , …, we have that implies that for all . This set of functions is said to be Linearly Dependent if where , , …, are not all zero for all .
Perhaps the simplest linearly independent sets of functions is that set that contains , , and . Let , , and be constants and consider the following equation:
It's not hard to see that equation above is satisfied if and only if the constants .
For another example, consider the functions and defined on all of . This set of functions is not linearly independent. To show this, let and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_2} be constants and consider the following equation:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad k_1f_1(t) + k_2f_2(t) = 0 \\ \quad k_1 \sin t + k_2 \sin (t + \pi) = 0 \end{align}}
Now choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \pi} . Then we have that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad k_1 \sin \pi + k_2 \sin 2\pi = 0 \\ \end{align}}
But the above equation is true for any choice of constants and since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \pi = \sin 2\pi = 0} , and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_2} do not form a linearly independent set on all of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} .
From the concept of linear independence/dependence, we obtain the following theorem on fundamental sets of solutions for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equations.
Theorem 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0} be an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equation. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_1(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_2(t)} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_n(t)} are solutions to this differential equation then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} form a fundamental set of solutions to this differential equation on the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly dependent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} .
- Proof: Consider the following Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equation:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0 \end{align}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow} Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_1(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_2(t)} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_n(t)} form a fundamental set of solutions to this differential equation on the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . Then this implies that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0 \in I} we have that :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad W(y_1, y_2, ..., y_n) \right|_{t_0} \neq 0 \end{align}}
- Thus this implies that following system of equations have only trivial solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = k_2 = ... = k_n = 0} :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0 \\ \quad k_1y_1'(t) + k_2y_2'(t) + ... + k_ny_n'(t) = 0 \\ \quad \quad \quad \quad \quad \quad \vdots \quad \quad \quad \quad \quad \quad \\ \quad k_1y_1^{(n-2)}(t) + k_2y_2^{(n-2)}(t) + ... + k_ny_n^{(n-2)}(t) = 0 \\ \quad k_1y_1^{(n-1)}(t) + k_2y_2^{(n-1)}(t) + ... + k_ny_n^{(n-1)}(t) = 0 \end{align}}
- Thus the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1y_1(t) + k_2y_2(t) + ... + k_ny_n(t) = 0} implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = k_2 = ... = k_n = 0} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly independent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} .
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow} We will prove the converse of Theorem 1 by contradiction. Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly independent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , and assume that instead Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} do NOT form a fundamental set of solutions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . Then for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0 \in I} , the Wronskian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(y_1, y_2, ..., y_n) \right|_{t_0} = 0} . Thus the system of equations above does not have only the trivial solution. Let the constants , , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_n^*} be a nontrivial solution to this system. Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t)} as:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \phi(t) = k_1^* y_1(t) + k_2^* y_2(t) + ... + k_n^* y_n(t) \end{align}}
- Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \phi(t)} satisfies the initial conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t_0) = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(t_0) = 0} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^{(n-1)} (t_0) = 0} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t)} satisfies our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equation because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t)} is a linear combination of the solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} .
- Now note that the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 0} also satisfies the differential equation and the initial conditions. By the existence/uniqueness theorem for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear homogenous differential equations, this implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t) = 0} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in I} , so:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 0 = k_1^* y_1(t) + k_2^* y_2(t) + ... + k_n^* y_n(t) \end{align}}
- But Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} are linearly independent which implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1^* = k_2^* = ... = k_n^* = 0} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1^*} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_2^*} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_n^*} is a trivial solution to the system above, which is a contradiction. Therefore our assumption that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n} do not form a fundamental set of solutions was false. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}
