CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q273 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Question 27 (Differential Equations)
Find the general solution of \(y^2\,dx+(x^2-xy+y^2)\,dy=0\). OR Solve \(dy/dx=y\tan x\) with \(y(0)=2\).
Alternative 1 3 Marks
Find the general solution of \(y^2\,dx+(x^2-xy+y^2)\,dy=0\).
Answer: tan⁻¹(x/y)+ln abs(y)=C on y≠0, and y=0.
- For y≠0, regard x as a function of y and put x=vy.
- Then dx/dy=−v²+v−1=v+yv′. Separate to obtain dv/(1+v²)=−dy/y.
- Integrate and substitute to get tan⁻¹(x/y)+ln abs(y)=C. The excluded constant branch y=0 satisfies the original differential equation.
Alternative 2 3 Marks
Solve \(dy/dx=y\tan x\) with \(y(0)=2\).
Answer: y=2 sec x for −π/2<x<π/2.
- Separate near the positive initial value to get dy/y=tan x dx.
- Integrate to get y=C sec x on the interval through zero.
- The initial condition gives C=2.