CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q293 Marksdifferential_equations
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 29 (Differential Equations)
Find the general solution of the differential equation x² dy/dx = x² + xy + y². OR Find the particular solution of the differential equation xy dy/dx = (x+2)(y+2), given that y(1) = −1.
Alternative 1 3 Marks
Find the general solution of the differential equation x² dy/dx = x² + xy + y².
Answer: tan⁻¹(y/x) = log|x| + C
- Divide by x²: dy/dx = 1 + y/x + (y/x)².
- Put y = vx ⇒ dy/dx = v + x dv/dx. v + x dv/dx = 1 + v + v² ⇒ x dv/dx = 1 + v² ⇒ dv/(1+v²) = dx/x. Integrate: tan⁻¹ v = log|x| + C ⇒ tan⁻¹(y/x) = log|x| + C.
- tan⁻¹(y/x) = log|x| + C
Alternative 2 3 Marks
Find the particular solution of the differential equation xy dy/dx = (x+2)(y+2), given that y(1) = −1.
Answer: y − 2 log|y+2| = x + 2 log|x| − 2
- Separate: y/(y+2) dy = (x+2)/x dx ⇒ (1 − 2/(y+2)) dy = (1+2/x) dx.
- Integrate: y − 2 log|y+2| = x + 2 log|x| + C. Using y(1) = −1: −1 − 0 = 1 + 0 + C ⇒ C = −2. Particular: y − 2 log|y+2| = x + 2 log|x| − 2.
- y − 2 log|y+2| = x + 2 log|x| − 2