Question

Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = e^x

Answer 1

We have,

\[y = e^x..........(1)\]

Differentiating both sides of (1) with respect to x, we get

\[\frac{dy}{dx} = e^x\]

⇒ \[\frac{dy}{dx} = y............\left[\text{Using (1)}\right]\]

It is the given differential equation.

Here, \[y = e^x\] satisfies the given differential equation; hence, it is a solution.

Also, when \[x = 0, y = e^0 = 1, i.e.,y(0) = 1 .\]

Hence, \[y = e^x\]  is the solution to the given initial value problem.