The solution of the differential equation x \frac{dy}{dx}+ 2y = x2 (x \neq 0) with y(1) = 1, is

Engineering

Mathematics

Linear Differential Equation

Question

The solution of the differential equation x $\frac{dy}{dx}$ + 2y = x² (x $\neq$ 0) with y(1) = 1, is

$y = \frac{3}{4} x^{2} + \frac{1}{4 x^{2}}$

$y = \frac{x^{2}}{4} + \frac{3}{4 x^{2}}$

$y = \frac{x^{3}}{5} + \frac{1}{5 x^{2}}$

$y = \frac{4}{5} x^{3} + \frac{1}{5 x^{2}}$

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

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

$\frac{dy}{dx} + (\frac{2}{x}) y = x (LDE in y)$

$IF = e^{\int \frac{2}{x} dx} = e^{2 ℓnx} = x^{2}$

$y . (x^{2}) = \int x . x^{2} dx = \frac{x^{4}}{4} + C$

y(1) = 1

$1 = \frac{1}{4} + C \Rightarrow C = 1 - \frac{1}{4} = \frac{3}{4}$

${yx}^{2} = \frac{x^{4}}{4} + \frac{3}{4}$

$y = \frac{x^{4}}{4} + \frac{3}{4 x^{2}}$