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UPSC 2017 Maths Optional Paper 1 Q7b-ii — Step-by-Step Solution

8 marks · Section B

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

Question

Solve the following differential equation using method of variation of parameters: $\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}-2y=44-76x-48x^2$.

Technique

$y_p=u_1y_1+u_2y_2$ with $u_1'=-y_2g/W,\ u_2'=y_1g/W$; integrate using $\int e^{kx}P\,dx$ shortcuts.

Solution

Step 1 — Complementary function

Auxiliary equation $m^2-m-2=0\Rightarrow(m-2)(m+1)=0\Rightarrow m=2,-1$:

Step 2 — Wronskian

Step 3 — Variation-of-parameters formulas

With $g(x)=44-76x-48x^2$, the particular solution is $y_p=u_1y_1+u_2y_2$ where

Step 4 — Evaluate the integrals

Using $\int e^{x}P(x)\,dx=e^{x}\big(P-P'+P''-\cdots\big)$ with $P=44-76x-48x^2$ ($P'=-76-96x,\ P''=-96$):

So $u_1=-\tfrac13 e^{x}(24+20x-48x^2)$, giving $u_1y_1=-\tfrac13(24+20x-48x^2)$.

For $u_2$, $\int e^{-2x}P\,dx=e^{-2x}\big(-\tfrac12P-\tfrac14P'-\tfrac18P''-\cdots\big)$. A direct computation (or sympy) gives

Adding $u_1y_1+u_2y_2$ and simplifying, the exponential parts cancel and the polynomial collapses to

(Indeed, since the forcing is a polynomial and $0$ is not a root of $m^2-m-2$, $y_p$ is a quadratic; variation of parameters reproduces it.)

Step 5 — General solution

Answer