Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #99
Solve x²y′ − xy = 1 + cos(y/x), x ≠ 0, y(1) = π/2.
✅ Step-by-Step Solution
Final Answer: tan(y/(2x)) = 3/2 − 1/(2x²); y = 2x arctan(3/2 − 1/(2x²)), x > 0.
- Put v = y/x. Since x²y′ − xy = x³v′, dv/(1 + cos v) = dx/x³ near v = π/2.
- Use 1 + cos v = 2cos²(v/2), giving ∫dv/(1 + cos v) = tan(v/2). Thus tan(v/2) = −1/(2x²) + C.
- At (1, π/2), tan(π/4) = 1, so C = 3/2.
- Thus y = 2x arctan(3/2 − 1/(2x²)), x > 0, with the branch through the initial point.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put v = y/x. Since x²y′ − xy = x³v′, dv/(1 + cos v) = dx/x³ near v = π/2.
- [1 mark] Use 1 + cos v = 2cos²(v/2), giving ∫dv/(1 + cos v) = tan(v/2). Thus tan(v/2) = −1/(2x²) + C.
- [1 mark] At (1, π/2), tan(π/4) = 1, so C = 3/2.
- [0.5 marks] Thus y = 2x arctan(3/2 − 1/(2x²)), x > 0, with the branch through the initial point.
Master Chapter 9: Differential Equations
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.
← Previous Question
Solve (x + 1)y′ − y = e^(3x)(x + 1)³.
Next Question →Solve y′ = xy/(x² + y²), 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?