Differential Equations
Homogeneous DE — Trigonometric Substitution
nta_pyq_2024_jan
Grade 12
Question:
If $\sin\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}$ is the solution of the differential equation $x\cos\left(\dfrac{y}{x}\right)\dfrac{dy}{dx}=y\cos\left(\dfrac{y}{x}\right)+x$ and $y(1)=\dfrac{\pi}{3}$, then $\alpha^2$ is equal to
Step-by-Step Solution
Key Concept: Let $v=y/x$; the DE becomes $\cos v\,dv=\frac{dx}{x}$. Integrate: $\sin v=\ln|x|+c$. Apply IC $y(1)=\pi/3$ to find $c=\alpha/2$.
Let $v=y/x$: $\cos v\,dv=\frac{dx}{x}\Rightarrow\sin v=\ln|x|+c$. IC: $c=\sin(\pi/3)=\frac{\sqrt{3}}{2}=\frac{\alpha}{2}\Rightarrow\alpha=\sqrt{3}$. $\alpha^2=3$.
Correct Answer: 1