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

Sep 8, 2016

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

#### Explanation:

$y ' = 2 x y$

separate it out

$\frac{1}{y} y ' = 2 x$

integrate

$\int \frac{1}{y} y ' \setminus \mathrm{dx} = \int 2 x \setminus \mathrm{dx}$

$\implies \int \frac{1}{y} \setminus \mathrm{dy} = \int 2 x \setminus \mathrm{dx}$

$\ln y = {x}^{2} + C q \quad \triangle$

$y = {e}^{{x}^{2} + C} = C {e}^{{x}^{2}}$ [C's just any constant]

apply the IV

$3 = C {e}^{{0}^{2}} \implies C = 3$

So

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

strictly speaking, $\triangle$ should be $\ln | y | = {x}^{2} + C$ with $| y | = C {e}^{{x}^{2}}$