Solutions of Linear Systems

Fundamental Solutions to 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^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 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^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0} with 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) = y_0} and $y'(t_{0})=y'_{0}$. 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 \phi} satisfies this differential equation. 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 W(y_1, y_2) \bigg|_{t_0} \neq 0} then we have that there exists constants $C$ 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} 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 y = Cy_1(t) + Dy_2(t)} satisfies this initial value problem. But since $p$ 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 q} are continuous 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} containing 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} then this implies that a unique solution exists, 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 \begin{align} \phi(t) = Cy_1(t) + Dy_2(t) \end{align}}

• So all solutions for this differential equation are 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 = 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 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} where the Wronskian 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 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} evaluated at 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} is nonzero. 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 y_0} and $y'_{0}$ be values for which the system 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 \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 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 $D$.

• Now 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 p} 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 q} are continuous on an open interval $I$ containing 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} , such a 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 \phi(t)} satisfies the initial conditions $y(t_{0})=y_{0}$ 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'(t_0) = y'_0} . Note though this solution is not a linear combination 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 y_1} and $y_{2}$ though which completes our proof. 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}

Theorem 1 above implies that if we can find two 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 = y_1(t)} and $y=y_{2}(t)$ for which 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) \neq 0} , then for 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 C} and $D$, all solutions of the second 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^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0} 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 \begin{align} \quad y = Cy_1(t) + Dy_2(t) \end{align}}

Also note that thus far we have not said that $y=y_{1}(t)$ 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 = y_2(t)} need to be distinct. However, with the Theorem above, we see that 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(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 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} .

Definition: 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^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0} where $p$ 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 q} are continuous on an 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} such that $t_{0}\in I$ and 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 y = y_1(t)} 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 = 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 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} 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 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)} for which 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)} is nonzero, 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 $y_{2}$ 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 $t_{0}\in I$. If $y=y_{1}(t)$ is a solution to this differential equation that satisfies the initial conditions $y_{1}(t_{0})=1$ and $y_{1}'(t_{0})=0$, and if $y=y_{2}(t)$ be a solution to this differential equation that satisfies the initial conditions $y_{2}(t_{0})=0$ and $y_{2}'(t_{0})=1$. Then $y_{1}$ and $y_{2}$ form a fundamental set of solutions for this differential equation.

• Proof: We note that:

${\begin{aligned}\quad W(y_{1},y_{2}){\bigg |}_{t_{0}}={\begin{vmatrix}y_{1}(t_{0})&y_{2}(t_{0})\\y_{1}'(t_{0})&y_{2}'(t_{0})\end{vmatrix}}={\begin{vmatrix}1&a0\\0&1\end{vmatrix}}=1\neq 0\end{aligned}}$

• Thus Theorem 1 implies that ALL solutions to this differential equation are given by $y=Cy_{1}(t)+Dy_{2}(t)$ where $C$ and $D$ are constants. Thus $y_{1}$ and $y_{2}$ form a fundamental set of solutions for this differential equation. $\blacksquare$

Licensing

Content obtained and/or adapted from:

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