# What is a solution to the differential equation dy/dx=12x^3y with the particular solution y(0)=2?

What is a solution to the differential equation $\frac{dy}{dx}=12{x}^{3}y$ with the particular solution y(0)=2?
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Kenny Kramer
This is a separable differential equation, so we can separate the variables and integrate:
$\frac{dy}{dx}=12{x}^{3}y$
$\frac{dy}{y}=12{x}^{3}dx$
$\int \frac{dy}{y}=12\int {x}^{3}dx$
$\mathrm{ln}|y|=3{x}^{4}+{C}_{1}$
$y\left(x\right)=C{e}^{3{x}^{4}}$
We can now determine the constant C from the initial condition:
$y\left(0\right)=2$
$2=C{e}^{0}=C$
so in conclusion:
$y\left(x\right)=2{e}^{3{x}^{4}}$