CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q303 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Question 30 (Differential Equations)
Solve (x²−y²)dx+2xy dy=0. OR Solve \(\sin x\cos y\,dx+\cos x\sin y\,dy=0\), with \(y(0)=\pi/4\).
Alternative 1 3 Marks
Solve (x²−y²)dx+2xy dy=0.
Answer: x²+y²=Cx on intervals with x≠0.
- Set u=x²+y², so u′=2x+2yy′.
- The given equation gives 2xyy′=y²−x², hence xu′=x²+y²=u.
- Thus d(u/x)/dx=0 for x≠0 and u=Cx.
Alternative 2 3 Marks
Solve \(\sin x\cos y\,dx+\cos x\sin y\,dy=0\), with \(y(0)=\pi/4\).
Answer: cos x cos y=1/√2 on the branch through (0,π/4).
- The left side is −d(cos x cos y).
- It follows that cos x cos y is constant on a solution.
- At the initial point the constant is 1·cos(π/4)=1/√2.