Question

solve the differential equation (seperable):
dx/dt -x^3 = x

Answer 1

seperating it is easy obviously. I'm tempted to use integration by parts but it doesn't really lead me anywhere

Answer 2

1/x(x^2 + 1) = A/x + (Bx + C)/(x^2 +1)

Answer 3

Another way to solve: Substituting \(y=x^{-2}\) gives \(y'=-2x^{-3}x'\), then
\[x'-x^3=x~~\implies~~x^{-3}x'-1=x^{-2}~~\implies~~-\frac{1}{2}y'-1=y\]
which is linear in \(y\) and also separable. Separating your variables yields
\[\frac{dy}{y+1}=-2\,dt\]