(1 point) The equation 3y′=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 (∗) as y′=xu′+u=f(u) where f(u)= 1/3(3u+2u^2+2) .

Separating variables we can write the equation in the form g(u)du=dxx where g(u)= 3/(2(u^2+1)) .

An implicit general solution with dependent variable u can be written in the form ln(x)− 3/2arctan(u) =C.

Transforming u=y/x back into the variables x and y and using the initial condition y(1)=1 we find C= -(3pi)/8

Finally solve for y to obtain the explicit solution of the initial value problem y= xtan(2/3lnx+(3pi)/8) Note: You can earn partial credit on this problem.