Let's begin by multiplying both sides of the equation by #r²#
#r^3 = 4r²theta - r²sin(theta) + r²cos²(theta)#
Because #x = rcos(theta)#, we can replace #r²cos²(theta)# with #x²#:
#r^3 = 4r²theta - r²sin(theta) + x²#
Because #y = rsin(theta)# and #r = sqrt(x² + y²)#, we can replace #- r²sin(theta)# with #-ysqrt(x² + y²)#:
#r^3 = 4r²theta - ysqrt(x² + y²) + x²#
The #4r²theta# term makes the one equation become 3 equations with 3 domain restrictions:
#r^3 = 4(x² + y²)(tan^-1(y/x)) - ysqrt(x² + y²) + x²; x > 0, y >= 0#
#r^3 = 4(x² + y²)(tan^-1(y/x) + pi) - ysqrt(x² + y²) + x²; x < 0#
#r^3 = 4(x² + y²)(tan^-1(y/x) + 2pi) - ysqrt(x² + y²) + x²; x > 0, y < 0#
Replace #r³# with #(x² + y²)^(3/2)#
#(x² + y²)^(3/2) = 4(x² + y²)(tan^-1(y/x)) - ysqrt(x² + y²) + x²; x > 0, y >= 0#
#(x² + y²)^(3/2)= 4(x² + y²)(tan^-1(y/x) + pi) - ysqrt(x² + y²) + x²; x < 0#
#(x² + y²)^(3/2) = 4(x² + y²)(tan^-1(y/x) + 2pi) - ysqrt(x² + y²) + x²; x > 0, y < 0#
