Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #97
Solve (x² + 3xy + y²)dx − x²dy = 0, y(1) = 0.
✅ Step-by-Step Solution
Final Answer: y = x ln x/(1 − ln x), 0 < x < e.
- For x ≠ 0 put v = y/x. Then v + xv′ = 1 + 3v + v², giving xv′ = (1 + v)².
- Near v = 0, integrate dv/(1 + v)² = dx/x to get −1/(1 + v) = ln|x| + C.
- At (1, 0), C = −1. Hence 1/(1 + v) = 1 − ln x.
- Therefore y = x ln x/(1 − ln x), on (0, e), the maximal interval through 1.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For x ≠ 0 put v = y/x. Then v + xv′ = 1 + 3v + v², giving xv′ = (1 + v)².
- [1 mark] Near v = 0, integrate dv/(1 + v)² = dx/x to get −1/(1 + v) = ln|x| + C.
- [1 mark] At (1, 0), C = −1. Hence 1/(1 + v) = 1 − ln x.
- [0.5 marks] Therefore y = x ln x/(1 − ln x), on (0, e), the maximal interval through 1.
Master Chapter 9: Differential Equations
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.
← Previous Question
Solve cos²x · y′ + y = tan x.
Next Question →Solve (x + 1)y′ − y = e^(3x)(x + 1)³.
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?