IGNOU MTE 8 2026 - Solve .b) Write the ordinary differential equation&nb

Question

Solve $\frac{dy}{dx} + xy = y^2 e^{x^2/2} \sin x$ .
b) Write the ordinary differential equation

$$y \, dx + (xy + x - 3y) \, dy = 0$

in the linear form, and hence find its solution.
c) Given that $y_1(x) = x^{-1}$ is one solution of the differential equation

$$2x^2y'' + 3xy' - y = 0, \quad x > 0,$

find a second linearly independent solution of the equation.

09 Jan 2026

Answer :

Word Count : 158

(a) [ \frac{dy}{dx}+xy=y^{2}e^{x^{2}/2}\sin x ] This is a Bernoulli equation. Divide by (y^{2}): [ y^{-2}\frac{dy}{dx}+x y^{-1}=e^{x^{2}/2}\sin x ] Let (v=y^{-1}). Then (\frac{dv}{dx}=-y^{-2}\frac{dy}{dx}). Hence [ -\frac{dv}{dx}+xv=e^{x^{2}/2}\sin x ] or [ \frac{dv}{dx}-xv=-e^{x^{2}/2}\sin x. ] This is linear in (v). The integrating factor is [ \mu=e^{\int(-x),dx}=e^{-x^{2}/2}. ] Multiplying, [ \frac{d}{dx}\left(ve^{-x^{2}/2}\right)=-\sin x. ] Integrating, [ ve^{-x^{2}/2}=\cos x+C. ] Thus [ v=e^{x^{2}/2}(\cos x+C).

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