Class 12 Differential Equations · Very Long Answer5 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #75
A solution y = f(x) of 2x²dy = (2xy + y²)dx passes through (1, 2). Find f(1/2).
✅ Step-by-Step Solution
Final Answer: f(1/2) = 1/(1 + ln 2).
- For x ≠ 0 put v = y/x, giving y′ = v + xv′.
- Then v + xv′ = v + v²/2, so 2dv/v² = dx/x near the nonzero initial value.
- Integrate −2/v = ln|x| + C, hence −2x/y = ln|x| + C.
- At (1, 2), C = −1; therefore y = 2x/(1 − ln x), 0 < x < e.
- At x = 1/2, f(1/2) = 1/(1 − ln(1/2)) = 1/(1 + ln 2).
📋 CBSE Stepwise Marking Scheme
- [1 mark] For x ≠ 0 put v = y/x, giving y′ = v + xv′.
- [1 mark] Then v + xv′ = v + v²/2, so 2dv/v² = dx/x near the nonzero initial value.
- [1 mark] Integrate −2/v = ln|x| + C, hence −2x/y = ln|x| + C.
- [1 mark] At (1, 2), C = −1; therefore y = 2x/(1 − ln x), 0 < x < e.
- [1 mark] At x = 1/2, f(1/2) = 1/(1 − ln(1/2)) = 1/(1 + ln 2).
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