Q. 4 - Calculus

Question

Differential Equations: A differential equation is simply an equation that involves derivatives. The solution of a differential equation is the family of functions that make the differential equation true.

• What family of functions satisfies the differential equation $f^{\text{'}}\left(x\right)=f\left(x\right)$ ? Notice that a function $f$ satisfying this differential equation has the property that it is its own derivative.

Step-by-Step Solution

Answer

Ans: Families of $e^x$ functions

Step 1. Given information:

The differential equation $f^{\text{'}}\left(x\right)=f\left(x\right)$

Step 2. Solving the differential equation :

By the definition of the derivative function,

$f^{\text{'}}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$

The differential equation $f^{\text{'}}\left(x\right)=f\left(x\right)$ is satisfied mostly by the families of the exponential function $e^x$.