Question

Verify $y=a{e}^x$ is a solution of $\frac{d^{2}y}{d{x}^2}-y=0$, find particular solution when $y\left(0\right)=1$.

Answer 1

We have,

$y=a{e}^x$ …… (1)

Differentiation this equation with respect to $x$ and we get,

$\frac{dy}{dx}=a{e}^x$

Again, differentiation this equation with respect to $x$ and we get,

$\frac{d^{2}y}{d{x}^2}=a{e}^x$ …… (2)

By equation (1) and (2) to, we get,

$\frac{d^{2}y}{d{x}^2}=y$

$\frac{d^{2}y}{d{x}^2}-y=0$

Hence proved.

Now, put $x=0$ in equation (1) and we get,

$y\left(0\right)=a{e}^0$

$y\left(0\right)=a$

Given that, $y\left(0\right)=1$

Then, $a=1$

Hence, this is the answer.