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UPSC 2014 Maths Optional Paper 1 Q6a — Step-by-Step Solution

10 marks · Section B

Method of variation of parameters · ODEs · asked 11× in 13 yrs · Read the full method →

Question

Solve by the method of variation of parameters:

Technique

Variation of parameters for first-order linear ODE; standard $\int e^{ax}\sin bx$ formula.

Solution

Strategy. For first-order linear ODE $y'+P(x)y=Q(x)$, variation of parameters replaces the integration constant $C$ in the homogeneous solution $y_h=Ce^{-\int P}$ by an unknown function $u(x)$.

Step 1 — Homogeneous solution

$y_h'=5y_h\Rightarrow y_h=Ce^{5x}$. So a fundamental solution is $y_1=e^{5x}$.

Step 2 — Variation: try $y_p=u(x)e^{5x}$

$y_p'=u'e^{5x}+5ue^{5x}$. Substitute into $y'-5y=\sin x$:

Step 3 — Integrate

Use the standard formula

With $a=-5,\,b=1$:

(Plus a constant of integration absorbed into the general solution.)

Step 4 — Particular and general solution

General solution:

Answer