CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q293 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Question 29 (Differential Equations)
Solve \(xy\prime=y-x\sin^2(y/x)\), given \(y(1)=\pi/6\). OR Solve \(y\ln y\,\frac{dx}{dy}+x=\frac2y\).
Alternative 1 3 Marks
Solve \(xy\prime=y-x\sin^2(y/x)\), given \(y(1)=\pi/6\).
Answer: cot(y/x)=ln x+√3 for x>0, with y/x in (0,π).
- Set u=y/x so y′=u+xu′. Substitute to get xu′=−sin²u on x>0.
- The initial value u(1)=π/6 has nonzero sine. Hence d(cot u)/dx=1/x, giving cot u=ln x+C.
- Apply cot(π/6)=√3 at x=1 to obtain C=√3 and select the branch through the initial value.
Alternative 2 3 Marks
Solve \(y\ln y\,\frac{dx}{dy}+x=\frac2y\).
Answer: x ln y=C−2/y, for y>0, with the continuous extension x(1)=2 when crossing y=1.
- Divide by y>0 and recognize the derivative of x ln y.
- The equation becomes d(x ln y)/dy=2/y². Integrate to obtain x ln y=−2/y+C.
- For y≠1 this determines x directly. A solution extending to y=1 must have C=2 and limit x(1)=2.