Differential Equations
Linear DE — Integrating Factor
nta_pyq_2024_jan
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $(1-x^2)\,dy=\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]\,dx$, $-1<x<1$, $y(0)=0$. If $y\left(\dfrac{1}{2}\right)=\dfrac{m}{n}$, $m$ and $n$ are coprime numbers, then $m+n$ is equal to
Step-by-Step Solution
Key Concept: Rewrite as $\frac{dy}{dx}-\frac{x}{1-x^2}y=\frac{(x^3+2)\sqrt{3}}{\sqrt{1-x^2}}$. I.F.$=e^{-\int\frac{x}{1-x^2}dx}=\sqrt{1-x^2}$. Integrate the RHS.
I.F.$=\sqrt{1-x^2}$: $y\sqrt{1-x^2}=\sqrt{3}\left(\frac{x^4}{4}+2x\right)$. $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}(\frac{1}{64}+1)}{\sqrt{1-\frac{1}{4}}}=\frac{\sqrt{3}\cdot\frac{65}{64}}{\frac{\sqrt{3}}{2}}=\frac{65}{32}$. $m+n=65+32=97$.
Correct Answer: 97