# What is the general solution of the differential equation? :  dy/dx=9x^2y

Jun 12, 2017

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

#### Explanation:

We have:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 9 {x}^{2} y$

This is a first Order linear Separable Differential Equation, we can collect terms by rearranging the equation as follows

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 9 {x}^{2}$

And now we can "separate the variables" to get

$\int \setminus \frac{1}{y} \setminus \mathrm{dy} = \int \setminus 9 {x}^{2} \setminus \mathrm{dx}$

And integrating gives us:

$\ln | y | = 9 {x}^{3} / 3 + C$

$\therefore \ln | y | = 3 {x}^{3} + C$

$\therefore | y | = {e}^{3 {x}^{3} + C}$

$\therefore | y | = {e}^{3 {x}^{3}} {e}^{C}$

And as ${e}^{x} > 0 \forall x \in \mathbb{R}$, we can write the solution as:

$\therefore y = A {e}^{3 {x}^{3}}$