Class 12 Differential Equations · Very Long Answer5 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #72
Show that (xe^(y/x) + y)dx = xdy is homogeneous and solve it with y(1) = 1.
✅ Step-by-Step Solution
Final Answer: e^(−y/x) + ln x = 1/e; y = −x ln(1/e − ln x), 0 < x < e^(1/e).
- Write y′ = e^(y/x) + y/x. This depends only on y/x and is homogeneous.
- Put v = y/x. Then v + xv′ = eᵛ + v, so e^(−v)dv = dx/x.
- Integrating gives −e^(−v) = ln|x| + C.
- At (1, 1), C = −1/e. Thus e^(−y/x) = 1/e − ln x.
- Therefore y = −x ln(1/e − ln x), on 0 < x < e^(1/e), where the logarithm argument is positive.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Write y′ = e^(y/x) + y/x. This depends only on y/x and is homogeneous.
- [1 mark] Put v = y/x. Then v + xv′ = eᵛ + v, so e^(−v)dv = dx/x.
- [1 mark] Integrating gives −e^(−v) = ln|x| + C.
- [1 mark] At (1, 1), C = −1/e. Thus e^(−y/x) = 1/e − ln x.
- [1 mark] Therefore y = −x ln(1/e − ln x), on 0 < x < e^(1/e), where the logarithm argument is positive.
Master Chapter 9: Differential Equations
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.
← Previous Question
Solve x dy − y dx = √(x² + y²)dx.
Next Question →Show that [x sin²(y/x) − y]dx + xdy = 0 is homogeneous and solve it…
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?