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UPSC 2016 Maths Optional Paper 1 Q5a — Step-by-Step Solution

10 marks · Section B

Particular integral via operator method · ODEs · asked 7× in 13 yrs · Read the full method →

Question

Find a particular integral of $\dfrac{d^2y}{dx^2}+y=e^{x/2}\sin\dfrac{x\sqrt3}{2}$.

Technique

Complex-exponential operator method, $y_p=\operatorname{Im}\dfrac{1}{D^2+1}e^{(\frac12+i\frac{\sqrt3}{2})x}$; substitute $D\to$ the complex exponent.

Solution

Write $D\equiv\dfrac{d}{dx}$. The equation is $(D^2+1)y=e^{x/2}\sin\dfrac{\sqrt3}{2}x$.

Step 1 — Check for resonance

The complementary function comes from $D^2+1=0\Rightarrow D=\pm i$, i.e. $\cos x,\sin x$. The forcing term $e^{x/2}\sin\frac{\sqrt3}{2}x$ is the imaginary part of $e^{(\frac12+i\frac{\sqrt3}{2})x}$, whose exponent $\frac12+i\frac{\sqrt3}{2}$ is not a root of $D^2+1$. Hence there is no resonance and the operator method applies directly.

Step 2 — Complex-exponential method

Take $y_p=\operatorname{Im}\,z_p$ where

Substitute $D\to\frac12+i\frac{\sqrt3}{2}$:

So

Step 3 — Rationalise the denominator

The denominator $\tfrac12+i\tfrac{\sqrt3}{2}=e^{i\pi/3}$ has modulus $1$. Multiply by its conjugate $\tfrac12-i\tfrac{\sqrt3}{2}$ (denominator $\tfrac14+\tfrac34=1$):

Step 4 — Take the imaginary part

Answer