Question

What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple$k$ of itself? Write each answer in the form of a first-order differential equation with a solution.

Short answer

If function $y=e^x$and its first derivative is $y\text{'}=e^x$.

If function $y=e^{kx}$and its first derivative is $y\text{'}=k{e}^{kx}$. so, A function that its first derivative is a constant multiple $k$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.

 First derivative.

Let we take the function,

$y=e^x$

Take first derivative,

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

So,$y=y\text{'}$

A function that its first derivative itself.

A function that its first derivative is a constant multiple$k$of itself.

$y=e^{kx}$

Take first derivative,

$y\text{'}=k{e}^{kx}$

Therefore,$y=e^x\to y\text{'}=e^x$and$y=e^{kx}\to y\text{'}=k{e}^{kx}$.