CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q303 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Question 30 (Differential Equations)
Solve \((y^2-x^2)\,dx=2xy\,dy\). OR Solve \((1+e^{2x})\,dy+(1+y^2)e^x\,dx=0\), with \(y(1)=0\).
Alternative 1 3 Marks
Solve \((y^2-x^2)\,dx=2xy\,dy\).
Answer: x²+y²=Cx on intervals with x≠0.
- Write 2xyy′=y²−x² and set u=x²+y².
- Then xu′=2x²+2xyy′=u, so (u/x)′=0 for x≠0.
- Thus u=Cx, giving the implicit family on its regular branches.
Alternative 2 3 Marks
Solve \((1+e^{2x})\,dy+(1+y^2)e^x\,dx=0\), with \(y(1)=0\).
Answer: y=(e−eˣ)/(1+e·eˣ)
- Separate to get dy/(1+y²)=−eˣdx/(1+e²ˣ).
- Integrate to obtain arctan y+arctan(eˣ)=C. The initial condition gives C=arctan e.
- Take the tangent of the difference to obtain y=(e−eˣ)/(1+e·eˣ).