Verify that$\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$,role="math" localid="1654838724304" $\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$role="math" localid="1654838779452" $\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{i}\mathbf{x}}$, and$\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{x}}$are all solutions of${\mathit{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{=}\mathbf{-}\mathit{y}$.

The given differential equation is$y^{\text{'}\text{'}}=-y$, and the solutions of the equation are$y=\sin x$,$y=\cos x$,$y=e^{ix}$, and$y=e^{ix}$.

In mathematics, an equation with only one independent variable and one or more of its derivatives with respect to the variable is referred to as an ordinarydifferentialequation, or ODE.In other words, the ODE is a relation with one independent variable x, a real dependent variable y, and several derivatives$y^{\text{'}},y^{\text{'}\text{'}},....,y^n$in relation to$x$.

Find the second derivative of the given solutions,and then substitute the solution and its second derivative into the differential equation.

For$y=\sin x$,

$\begin{array}{l}{y}^{\text{'}}=\cos x\\ y^{\text{'}\text{'}}=-\sin x\end{array}$

.

Substitute into the differential equation.

$-\sin x=-\sin x$

So$y=\sin x$, is a solution for the differential equation$y^{\text{'}\text{'}}=-y$.

For$y=\cos x$,

.$\begin{array}{l}{y}^{\text{'}}=-\sin x\\ y^{\text{'}\text{'}}=-\cos x\end{array}$

Substitute into the differential equation.

$-\cos x=-\cos x$

So$y=\cos x$, is a solution for the differential equation$y^{\text{'}\text{'}}=-y$.

For $y=e^{ix}$($y=e^{ix}$),

.$\begin{array}{l}{y}^{\text{'}}=i{e}^{ix}\\ y^{\text{'}\text{'}}=-e^{ix}\end{array}$

Substitute into the differential equation.

$-e^{ix}=-e^{ix}$

So$y=e^{ix}$, is a solution for the differential equation$y^{\text{'}\text{'}}=-y$.

For $y=-e^{-ix}$,

$\begin{array}{l}{y}^{\text{'}}=-i{e}^{-ix}\\ y^{\text{'}\text{'}}=e^{-ix}\end{array}$

Substitute in the differential equation.

$-e^{ix}=-e^{ix}$

So$y=e^{ix}$, is a solution for the differential equation$y^{\text{'}\text{'}}=-y$.

Hence, proved.