CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q273 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Question 27 (Differential Equations)
Solve (x²+y²)dy=xy dx. OR Solve \(y\prime-3y\cot x=\sin 2x\), with \(y(\pi/2)=2\).
Alternative 1 3 Marks
Solve (x²+y²)dy=xy dx.
Answer: x²/y²=2 ln |y|+C on nonzero branches. Also y=0.
- Locally take x as a function of y where xy≠0. Then 2x dx/dy=2(x²+y²)/y.
- Set u=x². The equation is u′−2u/y=2y, hence d(u/y²)/dy=2/y.
- Integrate to get x²/y²=2 ln |y|+C. The original equation also admits y=0.
Alternative 2 3 Marks
Solve \(y\prime-3y\cot x=\sin 2x\), with \(y(\pi/2)=2\).
Answer: y=4 sin³x−2 sin²x for 0<x<π.
- On the interval containing π/2, use the integrating factor sin⁻³x.
- Then d(y sin⁻³x)/dx=2 cos x/sin²x. Integration gives y sin⁻³x=−2/sin x+C.
- The initial condition gives C=4. Multiply through by sin³x.