Question

How do you tackle non-linear, non-separable, non-homogeneous ODEs? I.e.\[\frac{dy}{dx}=\sin(x+y)\]

Answer 1

Software churns out an enormous solution I'm befuddled by.

Answer 2

\[\frac{dy}{dx}=\sin(x+y)\]couldn't this be solved by doing something like:\[u=x+y\]\[\frac{du}{dx}=1+\frac{dy}{dx}\]so:\[\frac{dy}{dx}=\frac{du}{dx}-1\]substitue these into original equation to get:\[\frac{du}{dx}-1=\sin(u)\]\[\frac{du}{dx}=1+\sin(u)\]\[\int\frac{du}{1+\sin(u)}=\int{1dx}\]giving the solution:\[\frac{2\sin(\frac{u}{2})}{\sin(\frac{u}{2})+\cos(\frac{u}{2})}=x\]then substitute u=x+y to get final answer?