Question

Find the general solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=x^2$

Answer 1

Given differential equation :

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

Given differential equation is of the form

$\frac{dy}{dx}+Py=Q$

By comparing both the equations, we get

$P=\frac{1}{x}$ and $Q=x^2$

The general solution of the given differential equation is

$y(I.F)=\int (Q\times I.F.)dx+c$ ....(i)

Firstly, we need to find I.F.

$I.F.=e^{\int pdx}$

$I.F.=e^{\int \frac{dx}{x}}$

$I.F.=e^{\text{log}x}(\because e^{\text{log}x}=x)$

$I.F=x$

Substituting the value of I.F in (i), we get

$y\times x=\int x^2.xdx+c$

$xy=\int x^{3}dx+c$

$xy=\frac{x^4}{4}+c$

Hence, the required general solution is $xy=\frac{x^4}{4}+c$