Question

Solution of differential equation $\frac{dy}{dx}-2xy=x$ is

• A. $y=C{e}^{x^2}-\frac{1}{2}$
• B. $y=C{e}^{x^2}+\frac{1}{2}$
• C. $y=C{x}^2-\frac{1}{2}$
• D. None

Answer 1

Answer: (A)

$y=C{e}^{x^2}-\frac{1}{2}$

Given $\frac{dy}{dx}-2xy=x$ within in first order lines differential

equation of form $\frac{dy}{dx}+Py=Q$

$I.F=e^{\int Pdx}=e^{\int (-2x)dx}=C^{-x^2}$

Multiplying $I.F$ on both sides

$e^{-x^2}\frac{dy}{dx}-2xy.e^{-x^2}=x{e}^{-x^2}$

$d(e^{-x^2}.y)=x{e}^{-x^2}dx$

Integrating both sides

$e^{-x^2}y=\int x{e}^{-x^2}dx$

$=\frac{-e^{-x^2}}{2}+c$

$\Rightarrow y=c{e}^{x^2}-\frac{1}{2}$

$\therefore$ Option $A$ is correct