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:

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

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 $y_{2}$ 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. $\blacksquare$

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