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

Feb 14, 2017

$y \left(x\right) = 2 {e}^{3 {x}^{4}}$

#### Explanation:

This is a separable differential equation, so we can separate the variables and integrate:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 12 {x}^{3} y$

$\frac{\mathrm{dy}}{y} = 12 {x}^{3} \mathrm{dx}$

$\int \frac{\mathrm{dy}}{y} = 12 \int {x}^{3} \mathrm{dx}$

$\ln \left\mid y \right\mid = 3 {x}^{4} + {C}_{1}$

$y \left(x\right) = C {e}^{3 {x}^{4}}$

We can now determine the constant $C$ from the initial condition:

$y \left(0\right) = 2$

$2 = C {e}^{0} = C$

so in conclusion:

$y \left(x\right) = 2 {e}^{3 {x}^{4}}$