Class 12 Differential Equations · Long Answer3 Marksfirst_order_differential_equations
Class 12 Maths · Differential Equations · Long Answer #55
Solve (1 + e^(y/x))dy + e^(y/x)(1 − y/x)dx = 0, x ≠ 0.
✅ Step-by-Step Solution
Final Answer: y + xe^(y/x) = C, x ≠ 0.
- Compute d[xe^(y/x)] = e^(y/x)(1 − y/x)dx + e^(y/x)dy.
- The entire left-hand side is therefore d[y + xe^(y/x)].
- Integrating the exact differential gives y + xe^(y/x) = C. Its partial derivative with respect to y is 1 + e^(y/x) > 0.
- Thus y + xe^(y/x) = C defines local solution graphs on either side of x = 0, including the straight-line member when C = 0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Compute d[xe^(y/x)] = e^(y/x)(1 − y/x)dx + e^(y/x)dy.
- [1 mark] The entire left-hand side is therefore d[y + xe^(y/x)].
- [1 mark] Integrating the exact differential gives y + xe^(y/x) = C. Its partial derivative with respect to y is 1 + e^(y/x) > 0.
- [0.5 marks] Thus y + xe^(y/x) = C defines local solution graphs on either side of x = 0, including the straight-line member when C = 0.
Master Chapter 9: Differential Equations
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.
← Previous Question
Solve y′ + 2y tan x = sin x, y(π/3) = 0.
Next Question →Solve (1 + x²)y′ + y = e^(arctan x).
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?