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
$\frac{dy}{dx}=e^x$
$\Rightarrow$$\frac{dy}{dx}=y$ [Using (1)]
It is the given differential equation.
Here, $y=e^x$ satisfies the given differential equation; hence, it is a solution.
Also, when $x=0,y=e^0=1,\text{i.e.,}y\left(0\right)=1.$
Hence, $y=e^x$ is the solution to the given initial value problem.