Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #53
Solve (1 + x²)y′ + 2xy = √(x² + 4).
✅ Step-by-Step Solution
Final Answer: (1 + x²)y = x√(x² + 4)/2 + 2ln(x + √(x² + 4)) + C.
- Write the linear equation with P = 2x/(1 + x²), giving IF = 1 + x².
- Then [(1 + x²)y]′ = √(x² + 4).
- Use ∫√(x² + a²)dx = ½[x√(x² + a²) + a²ln|x + √(x² + a²)|], with a = 2.
- Thus (1 + x²)y = x√(x² + 4)/2 + 2ln(x + √(x² + 4)) + C, for all real x.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Write the linear equation with P = 2x/(1 + x²), giving IF = 1 + x².
- [1 mark] Then [(1 + x²)y]′ = √(x² + 4).
- [1 mark] Use ∫√(x² + a²)dx = ½[x√(x² + a²) + a²ln|x + √(x² + a²)|], with a = 2.
- [0.5 marks] Thus (1 + x²)y = x√(x² + 4)/2 + 2ln(x + √(x² + 4)) + C, for all real x.
Master Chapter 9: Differential Equations
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.
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?