MixedMediumQuestion 370011
Question
Solve: (1+x²) dy + 2xy dx = d(x² sin y)?
- AThis requires checking if exact: ∂M/∂y = ∂N/∂xCorrect
- BDirect integration
- CVariable separable directly
- DLinear ODE method
Correct answer
This requires checking if exact: ∂M/∂y = ∂N/∂x
Explanation
Write as M dx + N dy = 0 where M = 2xy, N = 1+x². Check exactness: ∂M/∂y = 2x, ∂N/∂x = 2x → exact DE. Solution: ∫M dx = x²y + g(y). Differentiate w.r.t. y: x² + g'(y) = N = 1+x² → g'(y) = 1 → g(y) = y. Solution: x²y + y = C → y(x²+1) = C.