The equation3y′=3xy+2x2+2y2x2, (∗)can be written in the form y′=f(y/x), i.e., it is homogeneous, so we can use the substitution
u=y/x to obtain a separable equation with dependent variable u=u(x). Introducing this substitution and using the fact that y′=xu′+u we can write (∗) asy′=xu′+u=f(u)where f(u)=Separating variables we can write the equation in the formg(u)du=dxxwhere g(u)= . An implicit general solution with dependent variable u can be written in the formln(x)−=CTransforming u=y/x back into the variables x and y and using the initial condition y(1)=1 we findC=Finally solve for y to obtain the explicit solution of the initial value problemy=
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