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UPSC 2022 Maths Optional Paper 1 Q5a — Step-by-Step Solution

10 marks · Section B

Linear first-order · ODEs · asked 5× in 13 yrs · Read the full method →

Question

Show that the general solution of $\dfrac{dy}{dx}+Py=Q$ can be written as

where $P,Q$ are non-zero functions of $x$ and $C$ is an arbitrary constant.

Technique

Standard integrating-factor solution; integration by parts rearranges the integral to extract $Q/P$ as an “equilibrium” term plus the homogeneous solution.

Solution

Setup. The standard solution of $y'+Py=Q$ uses integrating factor $\mu=e^{\int P\,dx}$:

We need to rewrite this as the given form.

Step 1 — Integrate by parts on $\int\mu Q\,dx$

In $\int e^{\int P\,dx}\cdot Q\,dx$, write $Q=P\cdot(Q/P)$. Then

Recognise $e^{\int P\,dx}P\,dx=d(e^{\int P\,dx})$ since $\dfrac{d}{dx}e^{\int P\,dx}=P e^{\int P\,dx}$.

So $\int e^{\int P\,dx}P\cdot\dfrac{Q}{P}\,dx=\int\dfrac{Q}{P}\,d(e^{\int P\,dx})$.

Step 2 — Apply integration by parts

$\int u\,dv=uv-\int v\,du$ with $u=Q/P$, $dv=d(e^{\int P\,dx})$:

Step 3 — Assemble

$\mu y=\int\mu Q\,dx+C_0=\dfrac{Q}{P}\mu-\int\mu\,d(Q/P)+C_0$.

Divide by $\mu=e^{\int P\,dx}$:

Set $C=-C_0$ (a different arbitrary constant):

Answer