Question

Solve the differential equation: $\frac{dy}{dx}=x^{2} y+y$

Answer 1

To solve the differential equation: $\frac{dy}{dx}=x^{2} y+y$,

separate the variables,i. e.taking x & dx on one side and y &dy on other

$\frac{dy}{dx}=x^{2} y+y \\ \\ \frac{dy}{dx} = y( {x}^{2} + 1) \\ \\ \frac{1}{y} dy = ( {x}^{2} + 1)dx \\ \\ integrate \: both \: sides \\\\\\\int\frac{1}{y} dy = \int( {x}^{2} + 1)dx\\\\\int\frac{1}{y} dy = \int{x}^{2} dx+\int 1.dx\\\\log \: y = \frac{ {x}^{3} }{3} + x + c$

is the solution.