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 $y_{2}$ 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}10\\01\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. $\blacksquare$

Fundamental sets of solutions

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

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

{\begin{aligned}\quad \left\{{\begin{matrix}x_{1}'=x_{1}\\x_{2}'=2x_{2}\end{matrix}}\right.\end{aligned}}

We can easily solve this system. For the first differential equation:

{\begin{aligned}\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_{1}e^{t}\end{aligned}}

Where $C_{1}=e^{C}>0$ .

For the second differential equation:

{\begin{aligned}\quad {\frac {dx_{2}}{dt}}&=2x_{2}\\{\frac {dx_{2}}{x_{2}}}&=2dt\\\int {\frac {1}{x_{2}}}dx_{2}&=\int 2dt\\\ln(x_{2})&=2t+C\\x_{2}&=e^{2t+C}\\x_{2}&=C_{2}e^{2t}\end{aligned}}

Where $C_{2}=e^{C}>0$ .

Note that in fact $C_{1},C_{2}$ can be any real numbers are we are not simply restricted to $C_{1},C_{2}>0$ .

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

We will now show that $\{\phi ^{[1]},\phi ^{[2]}\}$ is a Fundamental set of solutions to this system on all of $\mathbb {R}$ . Let $\alpha ,beta\in \mathbb {R}$ and consider the following equation:

{\begin{aligned}\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{aligned}}

The equation above implies that $\alpha e^{t}=0$ and $\beta e^{2t}=0$ for all $t\in \mathbb {R}$ . Since $e^{t},e^{2t}>0$ for all $t\in \mathbb {R}$ this implies that $\alpha ,\beta =0$ . So $\{\phi ^{[1]},\phi ^{[2]}\}$ is a linearly independent set of solutions to this system and so $\{\phi ^{[1]},\phi ^{[2]}\}$ is a fundamental set of solutions to this system on $\mathbb {R}$ .