Fundamental Solutions

Fundamental solutions to linear homogeneous equations

Theorem 1: Let ${\frac {d^{2}y}{dt^{2}}}+p(t){\frac {dy}{dt}}+q(t)y=0$ be a second order linear homogenous differential equation where $p$ and $q$ are continuous on an open interval $I$ such that $t_{0}\in I$, and let $y=y_{1}(t)$ and $y=y_{2}(t)$ be two solutions to this differential equation. The set of all linear combinations of these two solutions, $y=Cy_{1}(t)+Dy_{2}(t)$ where $C$ and $D$ are constants contains all solutions to this differential equation if and only if there exists a point $t_{0}$ for which the Wronksian of $y_{1}$ and $y_{2}$ at $t_{0}$ is nonzero, that is $W(y_{1},y_{2}){\bigg |}_{t_{0}}\neq 0$.

• Proof: Let $y=y_{1}(t)$ and $y=y_{2}(t)$ both be solutions to the differential equation ${\frac {d^{2}y}{dt^{2}}}+p(t){\frac {dy}{dt}}+q(t)y=0$ and suppose that $y=\phi (t)$ is any arbitrary solution as well. We want to show that $\phi (t)$ is a linear combination of $y_{1}$ and $y_{2}$ for some constants $C$ and $D$.

• $\Leftarrow$ Let $t_{0}$ be such that the Wronskian of $y_{1}$ and $y_{2}$ evaluated at $t_{0}$ is nonzero, that is:

${\begin{aligned}\quad W(y_{1},y_{2}){\bigg |}_{t_{0}}\neq 0\end{aligned}}$

• Take this value $t_{0}$ and evaluate both $\phi$ and $\phi '$ at this point. Then $\phi (t_{0})=y_{0}$ and $\phi '(t_{0})=y'_{0}$ (since $y=\phi (t)$ is a solution to our differential equation). Now consider the initial value problem ${\frac {d^{2}y}{dt^{2}}}+p(t){\frac {dy}{dt}}+q(t)y=0$ with the initial conditions $y(t_{0})=y_{0}$ and $y'(t_{0})=y'_{0}$. The function $\phi$ satisfies this differential equation. Since $W(y_{1},y_{2}){\bigg |}_{t_{0}}\neq 0$ then we have that there exists constants $C$ and $D$ such that $y=Cy_{1}(t)+Dy_{2}(t)$ satisfies this initial value problem. But since $p$ and $q$ are continuous on the open interval $I$ containing $t_{0}$ then this implies that a unique solution exists, and so:

${\begin{aligned}\phi (t)=Cy_{1}(t)+Dy_{2}(t)\end{aligned}}$

• So all solutions for this differential equation are a linear combination of the solutions $y=y_{1}(t)$ and $y=y_{2}(t)$.

• $\Rightarrow$ Suppose that every point $t_{0}\in I$ is such that $W(y_{1},y_{2}){\bigg |}_{t_{0}}=0$, that is, there exists no point $t_{0}$ on $I$ where the Wronskian of $y_{1}$ and $y_{2}$ evaluated at $t_{0}$ is nonzero. Let $y_{0}$ and $y'_{0}$ be values for which the system $\left\{{\begin{matrix}Cy_{1}(t_{0})+Dy_{2}(t_{0})=y_{0}\\Cy_{1}'(t_{0})+Dy_{2}'(t_{0})=y'_{0}\end{matrix}}\right.$ has no solutions for a set of constants $C$ and $D$.

• Now since $p$ and $q$ are continuous on an open interval $I$ containing $t_{0}$, such a solution $\phi (t)$ satisfies the initial conditions $y(t_{0})=y_{0}$ and $y'(t_{0})=y'_{0}$. Note though this solution is not a linear combination of $y_{1}$ and $y_{2}$ though which completes our proof. $\blacksquare$

Theorem 1 above implies that if we can find two solutions $y=y_{1}(t)$ and $y=y_{2}(t)$ for which the Wronskian $W(y_{1},y_{2})\neq 0$, then for constants $C$ and $D$, all solutions of the second order linear homogenous differential equation ${\frac {d^{2}y}{dt^{2}}}+p(t){\frac {dy}{dt}}+q(t)y=0$ are given by:

${\begin{aligned}\quad y=Cy_{1}(t)+Dy_{2}(t)\end{aligned}}$

Also note that thus far we have not said that $y=y_{1}(t)$ and $y=y_{2}(t)$ need to be distinct. However, with the Theorem above, we see that if $y_{1}(t)=y_{2}(t)$ then the Wronskian $W(y_{1},y_{2})=W(y_{1},y_{1})=W(y_{2},y_{2})$ is zero (as you should verify) and so not all solutions to a second order linear homogenous differential are given by the linear combination of just $y_{1}$.

Definition: Let ${\frac {d^{2}y}{dt^{2}}}+p(t){\frac {dy}{dt}}+q(t)y=0$ where $p$ and $q$ are continuous on an open interval $I$ such that $t_{0}\in I$ and let $y=y_{1}(t)$ and $y=y_{2}(t)$ be solutions to this differential equation. If the Wronskian $W(y_{1},y_{2})\neq 0$ then the set of linear combinations of $y_{1}$ and $y_{2}$ is known as the Fundamental Set of Solutions to this differential equation.

From the definition above, we see that if we can find two solutions $y=y_{1}(t)$ and $y=y_{2}(t)$ for which the Wronskian $W(y_{1},y_{2})$ is nonzero, then $y_{1}$ 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 y_2} form a fundamental set of solutions. The next question that we might pose is whether or not a second order linear homogenous differential equation always has a fundamental set of solutions.

Theorem 2: Let ${\frac {d^{2}y}{dt}}+p(t){\frac {dy}{dt}}+q(t)y=0$ be a second order linear homogenous differential equation where $p$ and $q$ are continuous on an open interval $I$ such 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 t_0 \in I} . 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)} is a solution to this differential equation that 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_1(t_0) = 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 y_1'(t_0) = 0} , and 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_2(t)} be a solution to this differential equation that 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_2(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 y_2'(t_0) = 1} . 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} 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 y_2} form a fundamental set of solutions for this differential equation.

• Proof: We 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 \begin{align} \quad W(y_1, y_2) \bigg|_{t_0} = \begin{vmatrix} y_1(t_0) y_2(t_0) \\ y_1'(t_0) y_2'(t_0)\end{vmatrix} = \begin{vmatrix} 1 0\\ 0 1 \end{vmatrix} = 1 \neq 0 \end{align}}

• Thus Theorem 1 implies that ALL solutions to this differential equation are given by 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 = Cy_1(t) + Dy_2(t)} where 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 C} 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 D} are constants. 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} 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 y_2} form a fundamental set of solutions for this 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 \blacksquare}

Fundamental sets of solutions

Definition: A Fundamental Set of Solutions to the linear homogeneous system of first order ODEs 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 \mathbf{x}' = A(t) \mathbf{x}} 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 J = (a, b)} is a set 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^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}} of linearly independent solutions to this system 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 J} .

For example, consider the following linear homogeneous system 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 2} first order ODEs:

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 \left\{\begin{matrix} x_1' = x_1\\ x_2' = 2x_2 \end{matrix}\right. \end{align}}

We can easily solve this system. For the first 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{dx_1}{dt} &= x_1 \\ \frac{dx_1}{x_1} &= dt \\ \int \frac{1}{x_1} dx &= \int dt \\ \ln (x_1) &= t + C \\ x_1 &= e^{t + C} \\ x_1 &= C_1e^t \end{align}}

Where 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 C_1 = e^C > 0} .

For the second 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{dx_2}{dt} &= 2x_2 \\ \frac{dx_2}{x_2} &= 2 dt \\ \int \frac{1}{x_2} dx_2 &= \int 2 dt \\ \ln (x_2) &= 2t + C \\ x_2 &= e^{2t + C} \\ x_2 &= C_2e^{2t} \end{align}}

Where 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 C_2 = e^C > 0} .

Note that in fact 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 C_1, C_2} can be any real numbers are we are not simply restricted to 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 C_1, C_2 > 0} .

Now by taking 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 C_1 = 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 C_2 = 0} we get 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^{[1]} = \begin{bmatrix} e^t\\ 0 \end{bmatrix}} is a solution to this system. Also, by taking 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 C_1 = 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 C_2 = 1} we get 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^{[2]} = \phi^{[1]} = \begin{bmatrix} 0\\ e^{2t} \end{bmatrix}} is a solution to this system.

We will now show 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^{[1]}, \phi^{[2]} \}} is a Fundamental set of solutions to this system 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}} . 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 \alpha, beta \in \mathbb{R}} 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} \alpha \phi^{[1]} + \beta \phi^{[2]} &= 0 \\ \alpha \begin{bmatrix} e^t\\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0\\ e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ \begin{bmatrix} \alpha e^t \\ \beta e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align}}

The equation above 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 \alpha e^t = 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 \beta e^{2t} = 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 \mathbb{R}} . 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 e^t, e^{2t} > 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 \mathbb{R}} 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 \alpha, \beta = 0} . 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 \{ \phi^{[1]}, \phi^{[2]} \}} is a linearly independent set of solutions to this system and 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 \{ \phi^{[1]}, \phi^{[2]} \}} is a fundamental set of solutions to this system 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 \mathbb{R}} .

Resources

• Fundamental Sets of Solutions Notes. Produced by Paul Dawkins, Lamar University
• Fundamental Set of Solutions, Brown University

Licensing

Content obtained and/or adapted from:

• Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com under a CC BY-SA license
• Fundamental Sets of Solutions, mathonline.wikidot.com under a CC BY-SA license