Question

Solve:

$\frac{dy}{dx}=(y+3x{)}^2$.

Answer 1

Given the differential equation,

$\frac{dy}{dx}=(y+3x{)}^2$......(1).

Let $y+3x=v$

or, $\frac{dy}{dx}+3=\frac{dv}{dx}$

or, $\frac{dy}{dx}=\frac{dv}{dx}-3$.

Using these in equation (1) we get,

$\frac{dv}{dx}-3=v^2$

or, $\frac{dv}{dx}=v^2+3$

or, $\frac{dv}{v^2+3}=dx$

Now integrating we have,

$\frac{1}{\sqrt{3}}{\tan }^1\frac{v}{\sqrt{3}}=x+c$ [ Where $c$ is integrating constant]

or, $\frac{1}{\sqrt{3}}{\tan }^1\frac{y+3x}{\sqrt{3}}=x+c$