Question

Solve the differential equation. (y + sin y)y' = x + x^3

Answer 1

$(y + \sin y)y'= x + x^3 \\ \Rightarrow (y + \sin y)\dfrac{dy}{dx}= x + x^3 \\ \Rightarrow (y + \sin y)dy= \left( x + x^3\right)dx \\ \Rightarrow \displaystyle\int(y + \sin y)dy= \int \left( x + x^3\right)dx \\ \\ \Rightarrow \dfrac{y^2}{2} - \cos y = \dfrac{x^2}{2} + \dfrac{x^4}{4} + C$

It is $\dfrac{y^2}{2} - \cos y = \dfrac{x^2}{2} + \dfrac{x^4}{4} + C$ as you cannot simplify further