Question

The solution of the differential equation ${\displaystyle x\frac{dy}{\text{d}x}+2y=x^2}$ is ______.

Answer 1

The solution of the differential equation ${\displaystyle x\frac{dy}{\text{d}x}+2y=x^2}$ is y = ${\displaystyle \frac{1}{4}{x}^2+\text{c}.x^{-2}}$.

Explanation:

The given differential equation is ${\displaystyle x\frac{dy}{\text{d}x}+2y=x^2}$

⇒ ${\displaystyle \frac{\text{d}y}{\text{d}x}+\frac{2}{x}y}$ = x.

Since, it is linear differential equation

∴ P = ${\displaystyle \frac{2}{x}}$ and Q = x

Integrating factor I.F. = ${\displaystyle {\text{e}}^{\int Pdx}}$

= ${\displaystyle {\text{e}}^{\int \frac{2}{x}\text{d}x}}$

= ${\displaystyle {\text{e}}^{2\log x}}$

= ${\displaystyle {\text{e}}^{{\log x}^2}}$

= x²

∴ Solution is ${\displaystyle y\times \text{I}.\text{F}.=\int \text{Q}\times \text{I}.\text{F}.\text{d}x+\text{c}}$

⇒ ${\displaystyle y.x^2=\int x.x^2\text{d}x+\text{c}}$

⇒ ${\displaystyle y.x^2=\int x^3\text{d}x+\text{c}}$

⇒ ${\displaystyle y.x^2=\frac{1}{4}{x}^4+\text{c}}$

⇒ y = ${\displaystyle \frac{1}{4}{x}^2+\text{c}.x^{-2}}$