Question

Show that xy^3-xy^3sin(x)= 1 is an implicit solution to the following

\[dy/dx = ((xcos(x) + \sin(x) - 1)y)/(3(x-xsin(x)))\]

Answer 1

\(\large\color{black}{ \displaystyle xy^3-xy^3\sin(x)= 1 }\)

Rearrangement:

\(\large\color{black}{ \displaystyle 1/y^3=x-x\sin(x) }\)

Calculating dy/dx:

\(\large\color{black}{ \displaystyle y'\times \left(-3/y^2\right)=1-(\sin(x) +x\cos(x))}\)

then rearrange it and solve for y'.