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UPSC 2018 Maths Optional Paper 1 Q7a — Step-by-Step Solution

13 marks · Section B

Euler-Cauchy equation · ODEs · asked 8× in 13 yrs · Read the full method →

Question

Solve: $(1+x)^2y''+(1+x)y'+y=4\cos(\log(1+x))$.

Technique

Cauchy–Euler substitution $t=\log(1+x)$ reduces to $(D^2+1)y=4\cos t$; resonant PI $\frac1{D^2+1}\cos t=\frac t2\sin t$.

Solution

This is a Cauchy–Euler (equidimensional) equation in the variable $(1+x)$.

Step 1 — Substitution

Let $t=\log(1+x)$, so $1+x=e^{t}$ and $\dfrac{dt}{dx}=\dfrac{1}{1+x}$. Writing $D=\dfrac{d}{dt}$, the standard reductions are

Step 2 — Transform the equation

A constant-coefficient equation in $t$.

Step 3 — Complementary function

$m^2+1=0\Rightarrow m=\pm i$:

Step 4 — Particular integral (resonance)

The forcing $4\cos t$ has frequency matching the roots $\pm i$, so it resonates. Using $\dfrac{1}{D^2+1}\cos t=\dfrac{t}{2}\sin t$ (the standard resonant result):

Step 5 — Back-substitute $t=\log(1+x)$

Answer