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)(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 (y^2)/(2) - \cos y = (x^2)/(2) + (x^4)/(4) + C`

It is
`(y^2)/(2) - \cos y = (x^2)/(2) + (x^4)/(4) + C` as you cannot simplify further