Question

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${\mathit{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{=}\mathbf{-}\mathit{y}$.

Short answer

It is verified that$y=\sin x$,role="math" localid="1654839035165" $y=cosx$,$y=e^{ix}$ , and $y=e^{-ix}$are the solutions for the differential equation$y^{\text{'}\text{'}}=-y$.

Step-by-step solution

 Step 1: Given information

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$y=e^{ix}$.

Meaning of differential equation

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$x$.

Verify thaty=sinx,y=cosx,y=eix, andsrc="data:image/svg+xml;base64,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" role="math" localid="1654840814053" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/20ce386b-8e95-4dde-a606-20e2fc264f01.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220610%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220610T064344Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=df7852048cd97ec5df031ec1e02cdaab3305f428a0605792fc55093d0b6eadb6" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/20ce386b-8e95-4dde-a606-20e2fc264f01.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220610%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220610T064143Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=c35cb4067fefa60e305387686a6a0b8ca1b1a10c4bb5214fc5a1533d330d5a50" y=e−ixare the solutions for the differential equationsrc="data:image/svg+xml;base64,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" role="math" localid="1654840880252" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/e1ee4441-0f76-49eb-8052-0fde77b8c58a.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220610%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220610T064344Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=dade9b9a38886bdbccf65c2fb27990f8d3d59f45e99f44a51affc84f97214cd4" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/e1ee4441-0f76-49eb-8052-0fde77b8c58a.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220610%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220610T064143Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=7a13e784436605b1d772c53c7a6c1a6ea075f1220607d7967736bfb4618c1cf5" y''=−y

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.