Question

What function (or functions) do you know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a second-order differential equation with a solution. Functions whose second derivative is itself.

Short answer

The function$y=e^x$makes its second derivative itself.

The function$y=\sin x$makes its second derivative is the negative of itself.

Step-by-step solution

Step 1:Definition of differential equation.

A differential equation is defined as the derivative function of one or more unknown functions (dependable variables) with respect to one or more undependable variables.

The function makes its second derivative itself.

Let we take the function,$y=e^x$

Take first derivative,

$y\text{'}=e^x$

Take second derivative,

$y"=e^x$

So,$y=y"$

If,$y=e^{-x}$

Take first derivative,

$y\text{'}=-e^{-x}$

Take second derivative,

$y"=e^{-x}$

So, $y=y"$

Functions whose second derivative is negative of itself.

Let we take the function,$y=\sin x$

Take first derivative,

$y\text{'}=\cos x$

Take second derivative,

$y"=-\sin x$

So,$y"=-y$

If,$y=\cos x$

Take first derivative,

$y\text{'}=-\sin x$

Take second derivative,

$y"=-\cos x$

So,$y"=-y$

Therefore, The function $y=e^x$ makes its second derivative itself and a function $y=\sin x$makes its second derivative is the negative of itself.