IGNOU BMTC 132 2026 - in is a linear homogeneous equation

Question

$\sin x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$ in $]0, \pi[$ is a linear homogeneous equation

09 Jan 2026

Answer :

Word Count : 457

We are asked to solve the linear homogeneous differential equation: [ \sin x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0, \quad x \in (0, \pi). ] First, rewrite it in standard form by dividing through by (\sin x) (valid for (x \in (0, \pi)), since (\sin x \neq 0)): [ \frac{d^2y}{dx^2} + \frac{1}{\sin x} \frac{dy}{dx} + \frac{1}{\sin x} y = 0. ] This is a second-order linear equation with variable coefficients. To attempt a solution, try a substitution inspired by reduction of order: let (y = e^{\lambda x}). Then: [ \frac{dy}{dx} = \lambda e^{\lambda x}, \quad \frac{d^2y}{dx^2} = \lambda^2 e^{\lambda x}. ] Substitute into the equation: [ \sin x (\lambda^2 e^{\lambda x}) + (\lambda e^{\lambda x}) + e^{\lambda x} = e^{\lambda x} (\lambda^2 \sin x + \lambda + 1) = 0. ] This gives: [ \lambda^2 \sin x + \lambda + 1 = 0, ] which depends on (x), so (y = e^{\lambda x}) is

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