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UPSC 2016 Maths Optional Paper 1 Q6c — Step-by-Step Solution

15 marks · Section B

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

Question

Find the general solution of the equation $x^2\dfrac{d^3y}{dx^3}-4x\dfrac{d^2y}{dx^2}+6\dfrac{dy}{dx}=4$.

Technique

Substitute $p=y'$ to drop to a 2nd-order Cauchy–Euler ($x^2p''-4xp'+6p=4$); indicial roots $2,3$; constant PI $p=2/3$; integrate once.

Solution

The lowest-order derivative present is $y'$, so the equation is really second order in $p=y'$.

Step 1 — Reduce the order

Let $p=\dfrac{dy}{dx}$. Then $y''=p'$, $y'''=p''$, and the equation becomes

This is a Cauchy–Euler (equidimensional) equation in $p$.

Step 2 — Solve the homogeneous Cauchy–Euler part

Put $x=e^{t}$, $\theta\equiv\dfrac{d}{dt}$, so $x p'=\theta p$ and $x^2 p''=\theta(\theta-1)p$. The homogeneous equation $x^2p''-4xp'+6p=0$ transforms to

Roots $m=2,3$, so $p_c=Ax^2+Bx^3$.

Step 3 — Particular solution of (1)

The RHS is the constant $4$. Try $p=k$ (constant): then $x^2\cdot0-4x\cdot0+6k=6k=4\Rightarrow k=\dfrac23$. So

Step 4 — Integrate to recover $y$

Renaming the arbitrary constants ($c_1=C,\ c_2=A/3,\ c_3=B/4$):

Answer