Solutions of Linear Systems

Fundamental Solutions to Linear Homogenous Differential 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:

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 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)} 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 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) = 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 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 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 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} 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 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 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 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)} 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 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 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 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 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 &a 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}

Licensing

Content obtained and/or adapted from:

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