Class 12 Differential Equations · Long Answer3 Marksseparable_differential_equations
Class 12 Maths · Differential Equations · Long Answer #106
Solve ln(y′) = 3x + 4y, y(0) = 0.
✅ Step-by-Step Solution
Final Answer: 4e^(3x) + 3e^(−4y) = 7; y = −¼ln((7 − 4e^(3x))/3), x < ⅓ln(7/4).
- Exponentiate and separate e^(−4y)dy = e^(3x)dx.
- Integrate −e^(−4y)/4 = e^(3x)/3 + C, or 4e^(3x) + 3e^(−4y) = C₁.
- At (0, 0), C₁ = 7.
- Thus y = −¼ln((7 − 4e^(3x))/3), x < ⅓ln(7/4). Its derivative is positive, as required by ln(y′).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Exponentiate and separate e^(−4y)dy = e^(3x)dx.
- [1 mark] Integrate −e^(−4y)/4 = e^(3x)/3 + C, or 4e^(3x) + 3e^(−4y) = C₁.
- [1 mark] At (0, 0), C₁ = 7.
- [0.5 marks] Thus y = −¼ln((7 − 4e^(3x))/3), x < ⅓ln(7/4). Its derivative is positive, as required by ln(y′).
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