# How do you solve the differential equation given f'(x)=4x, f(0)=6?

Oct 31, 2016

$y = f \left(x\right) = 2 {x}^{2} + 6$

#### Explanation:

Substitute $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $f ' \left(x\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 4 x$

Use the separation of variable technique:

$\mathrm{dy} = 4 x \mathrm{dx}$

Integrate both sides:

$y = 2 {x}^{2} + C$

Evaluate C, using the initial condition:

$6 = 2 {\left(0\right)}^{2} + C$

$C = 6$