Differential Equations Question 13
Question 13 - 2024 (30 Jan Shift 1)
Let $y=y(x)$ be the solution of the differential equation $\left(1-x^{2}\right) d y=\left[x y+\left(x^{3}+2\right) \sqrt{3\left(1-x^{2}\right)}\right] d x$, $-1<x<1, y(0)=0$. If $y\left(\frac{1}{2}\right)=\frac{m}{n}, m$ and $n$ are coprime numbers, then $m+n$ is equal to
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Answer (97)
Solution
$\frac{d y}{d x}-\frac{x y}{1-x^{2}}=\frac{\left(x^{3}+2\right) \sqrt{3\left(1-x^{2}\right)}}{1-x^{2}}$
$IF=e^{-\int \frac{x}{1-x^{2}} dx}=e^{+\frac{1}{2} \ln \left(1-x^{2}\right)}=\sqrt{1-x^{2}}$
$y \sqrt{1-x^{2}}=\sqrt{3} \int\left(x^{3}+2\right) d x$
$y \sqrt{1-x^{2}}=\sqrt{3}\left(\frac{x^{4}}{4}+2 x\right)+c$
$\Rightarrow y(0)=0 \quad \therefore c=0$
$y\left(\frac{1}{2}\right)=\frac{65}{32}=\frac{m}{n}$
$m+n=97$
