Class 12 Differential Equations · Long Answer3 Marksseparable_differential_equations
Class 12 Maths · Differential Equations · Long Answer #60
Solve y′ = 1 + x + y + xy, y(1) = 0.
✅ Step-by-Step Solution
Final Answer: y = exp(x + x²/2 − 3/2) − 1.
- Factor the right-hand side: y′ = (1 + x)(1 + y).
- Near y = 0, integrate dy/(1 + y) = (1 + x)dx to get ln|1 + y| = x + x²/2 + C.
- At (1, 0), C = −3/2 and 1 + y is positive.
- Therefore y = exp(x + x²/2 − 3/2) − 1, for all real x.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Factor the right-hand side: y′ = (1 + x)(1 + y).
- [1 mark] Near y = 0, integrate dy/(1 + y) = (1 + x)dx to get ln|1 + y| = x + x²/2 + C.
- [1 mark] At (1, 0), C = −3/2 and 1 + y is positive.
- [0.5 marks] Therefore y = exp(x + x²/2 − 3/2) − 1, for all real x.
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