Question

The solution of the differential equation $\frac{dy}{dx}+2xy=e^{-x^2}$ with y(0) = 1 is

• A. (1 + x) $e^{+x^2}$
• B. (1 + x) $e^{-x^2}$
• C. (1 - x) $e^{+x^2}$
• D. (1 - x) $e^{-x^2}$

Answer 1

The correct option is B (1 + x) $e^{-x^2}$

$\frac{dy}{dx}+2xy=e^{-x^2};y\left(0\right)=1$

Integrating factor,

I.F. = $e^{\int \left(2x\right)dx}=e^{x^2}$

General solution,

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

y $e^{x^2}=x+c$

y = (x + c) $e^{-x^2}$

y(0) = 1

$\Rightarrow 1=(0+c)e^{-0}$

$\Rightarrow c=1$

y = (x + 1) $e^{-x^2}$