Differential Equations
Homogeneous DE — Substitution y = vx
nta_pyq_2026_jan
Grade 12
Question:
Let the solution curve of the differential equation $x\,dy-y\,dx=\sqrt{x^2+y^2}\,dx$, $x>0$, $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to
Step-by-Step Solution
Key Concept: Substitute $y=vx$: $\dfrac{dv}{\sqrt{1+v^2}}=\dfrac{dx}{x}$. Integrate: $\ln(v+\sqrt{1+v^2})=\ln x+C$.
$y+\sqrt{x^2+y^2}=x^2$. $y(3)=4$.
Correct Answer: 2