Find the unique function $\displaystyle y(x)$ satisfying the differential equation with initial condition,
$\displaystyle \frac{dy}{dx}=x^2 y$, $\displaystyle y(1)=1$
I'm guessing I have to solve for y in terms of x, let y = 1 and x = 1, then solve for C?
This is what I did:
$\displaystyle y = e^\frac{x^3}{3} + C$
Since $\displaystyle y(1)=1$,
$\displaystyle 1=e^\frac{1}{3} + C$
$\displaystyle C=1 - e^\frac{1}{3}$
For some reason this doesn't feel right.
Then the original equation should be:
$\displaystyle y = e^\frac{x^3}{3} + 1 - e^\frac{1}{3}$