Differential Equations
Exact DE / Separable — Evaluating Parameters
nta_pyq_2024_jan
Grade 12
Question:
If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x)\,dx+x\,dy=0$, $x>0$ passes through the point $(1,1)$ and $y(e)=\dfrac{\alpha-\tan\left(\frac{3}{2}\right)}{\beta+\tan\left(\frac{3}{2}\right)}$, then $\alpha+2\beta$ is
Step-by-Step Solution
Key Concept: Separate: $\frac{dy}{1+y^2}=-\frac{(1+\ln x)}{x}dx$. Integrate both sides. Apply IC $(1,1)$ to find $C$. Then substitute $x=e$ to get $y(e)$ and match to the given form to extract $\alpha,\beta$.
$\tan^{-1}y+\frac{(\ln x)^2}{2}+\ln x=C$. IC: $C=\frac{\pi}{4}$. At $x=e$: $\tan^{-1}y=\frac{\pi}{4}-\frac{3}{2}\Rightarrow y=\frac{1-\tan(3/2)}{1+\tan(3/2)}$. Matching: $\alpha=1,\beta=1$, $\alpha+2\beta=3$.
Correct Answer: 3