Question

The differential equation $x\frac{dy}{dx}-y=x^2$, has the general solution
(a) y − x³ = 2cx
(b) 2y − x³ = cx
(c) 2y + x² = 2cx
(d) y + x² = 2cx

Answer 1

(b) 2y − x³ = cx


We have,

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

$$\begin{gathered} \Rightarrow \frac{dy}{dx}-\frac{1}{x}y=x^2 \\ \mathrm{Comparing}\mathrm{with}\frac{dy}{dx}+Py=Q,\mathrm{we}\mathrm{get} \\ P=-\frac{1}{x} \\ Q=x^2 \\ \mathrm{Now}, \\ \mathrm{I}.\mathrm{F}.=e^{-\int \frac{1}{x}dx}=e^{-\log \left|x\right|} \\ =e^{\log \left|\frac{1}{x}\right|} \\ =\frac{1}{x} \\ y\times \mathrm{I}.\mathrm{F}=\int x^{\mathit{2}}\times \mathrm{I}.\mathrm{Fd}x+C \\ \Rightarrow y\frac{1}{x}=\int x^2\times \frac{1}{x}dx+C \\ \Rightarrow y\frac{1}{x}=\int xdx+C \\ \Rightarrow y\frac{1}{x}=\frac{x^2}{2}+C \\ \Rightarrow 2y-x^3=Cx \end{gathered}$$