Class 12 Differential Equations · Very Long Answer5 Marksseparable_differential_equations

Class 12 Maths · Differential Equations · Very Long Answer #74

Solve y′ = x(2ln x + 1)/(sin y + y cos y), y(1) = π/2.

✅ Step-by-Step Solution

Final Answer: y sin y = x²ln x + π/2, on the regular branch through (1, π/2).

The equation requires x > 0 and sin y + y cos y ≠ 0. Separate (sin y + y cos y)dy = x(2ln x + 1)dx.
The left-hand side is d(y sin y).
The right-hand side is d(x²ln x), so integration gives y sin y = x²ln x + C.
At x = 1, y = π/2 gives C = π/2.
Thus y sin y = x²ln x + π/2 on the local regular branch through the initial point, retaining sin y + y cos y ≠ 0.

[1 mark] The equation requires x > 0 and sin y + y cos y ≠ 0. Separate (sin y + y cos y)dy = x(2ln x + 1)dx.
[1 mark] The left-hand side is d(y sin y).
[1 mark] The right-hand side is d(x²ln x), so integration gives y sin y = x²ln x + C.
[1 mark] At x = 1, y = π/2 gives C = π/2.
[1 mark] Thus y sin y = x²ln x + π/2 on the local regular branch through the initial point, retaining sin y + y cos y ≠ 0.

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Differential Equations.