Find the particular solution of the differential equation $\frac{dy}{dx} - 2xy = 3x^2 e^{x^2}$; $y(0) = 5$.

$y = (x^3 + 5)e^{x^2}$

The correct answer is Option (2) → $y = (x^3 + 5)e^{x^2}$ ##

Given differential equation is

$\frac{dy}{dx} - 2xy = 3x^2 e^{x^2}$

On comparing the above equation with $\frac{dy}{dx} + Py = Q$,

We get $P = -2x, Q = 3x^2 e^{x^2}$

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

$= e^{-\left( \frac{2x^2}{2} \right)} = e^{-x^2}$

$y \cdot e^{-x^2} = \int 3x^2 e^{x^2} \cdot (e^{-x^2}) dx + C$

or $y \cdot e^{-x^2} = 3 \int x^2 dx + C$

$\frac{y}{e^{x^2}} = 3 \left[ \frac{x^3}{3} \right] + C$

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

$y = e^{x^2} x^3 + C e^{x^2}$

Given $y(0) = 5$

$5 = 0 + C e^0$

$⇒C = 5$

Thus, the required solution is

$y = e^{x^2} (x^3 + 5)$