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Mathematics · 2022 · 3 marks
Differential EquationsMethods of Solving First Order, First Degree Differential Equations3 marksApplyshort answer
CBSE 2022 · Region 1 · Set 1 · Q10
Find the particular solution of the differential equation x dy/dx+x cos^2(y/x)= y; given that when x=1, y =(π)/4.
Find the particular solution of the differential equation $\displaystyle x \frac{\mathrm{dy}}{\mathrm{dx}}+x \cos ^{2}\left(\frac{\mathrm{y}}{x}\right)=\mathrm{y}$; given that when $\displaystyle x=1, \mathrm{y}=\frac{\pi}{4}$.
Marking-scheme solution
Given differential equation can be written as
\[\frac{d y}{d x}=\frac{y}{x}-\cos ^{2} \frac{y}{x}
\]
Let \(\displaystyle y=v x \therefore \frac{d y}{d x}=v+x \frac{d v}{d x}\)
(1)
Becomes \(\displaystyle \gamma+x \frac{d v}{d x}=\gamma-\cos ^{2} v\)
\[\sec ^{2} v d v=-\frac{d x}{x}
\]
Integrating both sides we get
\[\begin{aligned}
& \tan v=-\log |x|+c \\
& \tan \frac{y}{x}=-\log |x|+c
\end{aligned}
\]
\[x=1, y=\frac{\pi}{4} \Rightarrow c=1
\]
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CBSE Class 12 Mathematics past-paper question from the 2022 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.