Solve the IVP $\frac{dy}{dx}=-6xy$, $y(0)=7$

Question

$\frac{dy}{dx}=-6xy$, $y(0)=7$

First I separated the equations and got:

$\frac{1}{y}\,dy=-6x\,dx$

I then integrated both sides:

$\int\frac{1}{y}\,dy=\int-6x\,dx$

After solving I got:

$\ln|y|=-3x^2+C$

And found the solution:

$y=e^{3x^2}e^C$

I then plugged my initial value into the solution:

$7=e^{3(0)^2}e^C$

$7=e^C$

$\ln|7|=C$

So then I found the particular solution to be:

$y=7e^{3x^2}$

This is my first attempt at a separable diff'q and just want to see if I did it correctly.

Answer 1

by separation of unknows, we obtain $$y(x)=Cexp(-3x^2)$$ and $$y(0)=7$$ gives $$C=7$$