Class 12 Differential Equations · Very Long Answer5 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #111
Solve dx/dy + x cot y = 2y + y²cot y, y ≠ 0, with x = 0 when y = π/2.
✅ Step-by-Step Solution
Final Answer: x sin y = y²sin y − π²/4, on 0 < y < π.
- This is linear in x(y), with P = cot y. A convenient IF is sin y.
- Then d(x sin y)/dy = 2y sin y + y²cos y.
- The right side is d(y²sin y)/dy, so x sin y = y²sin y + C.
- At y = π/2 and x = 0, C = −π²/4.
- Thus x sin y = y²sin y − π²/4, on (0, π), where cot y is defined.
📋 CBSE Stepwise Marking Scheme
- [1 mark] This is linear in x(y), with P = cot y. A convenient IF is sin y.
- [1 mark] Then d(x sin y)/dy = 2y sin y + y²cos y.
- [1 mark] The right side is d(y²sin y)/dy, so x sin y = y²sin y + C.
- [1 mark] At y = π/2 and x = 0, C = −π²/4.
- [1 mark] Thus x sin y = y²sin y − π²/4, on (0, π), where cot y is defined.
Master Chapter 9: Differential Equations
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.
← Previous Question
Solve xy y′ = (x + 2)(y + 2), y(1) = −1.
Next Question →Solve x(1 + y²)dx − y(1 + x²)dy = 0, y(0) = 1.
More Practice Questions in Differential Equations
Related Questions in Differential Equations
Class 12 Differential Equations · Mcq Single Correct
Find the degree of x²y″ = (xy′ − y)³.
Class 12 Differential Equations · Mcq Single Correct
Find the degree of y″ + 3(y′)² = x² ln(y″).
Class 12 Differential Equations · Mcq Single Correct
Find the order and degree of [1 + (y′)²]² = y″.
Class 12 Differential Equations · Mcq Single Correct
The nonconstant solutions of 2xy′ − y = 3 lie on a family of which curves?
Class 12 Differential Equations · Mcq Single Correct
Find an integrating factor of (x ln x)y′ + y = 2 ln x, for x > 0, x ≠ 1.
Class 12 Differential Equations · Mcq Single Correct
Which function satisfies (y′)² − xy′ + y = 0?