4.38.36 \(a x y(x)+b-\left (k x^{k-1}-12 x^2\right ) \left (3 y'(x)+y(x)^2\right )+2 \left (x^k-4 x^3\right ) \left (y''(x)+y(x) y'(x)-y(x)^3\right )=0\)

ODE
\[ a x y(x)+b-\left (k x^{k-1}-12 x^2\right ) \left (3 y'(x)+y(x)^2\right )+2 \left (x^k-4 x^3\right ) \left (y''(x)+y(x) y'(x)-y(x)^3\right )=0 \] ODE Classification

[NONE]

Book solution method
TO DO

Mathematica
cpu = 5.158 (sec), leaf count = 0 , could not solve

DSolve[b + a*x*y[x] - (-12*x^2 + k*x^(-1 + k))*(y[x]^2 + 3*Derivative[1][y][x]) + 2*(-4*x^3 + x^k)*(-y[x]^3 + y[x]*Derivative[1][y][x] + Derivative[2][y][x]) == 0, y[x], x]

Maple
cpu = 3.817 (sec), leaf count = 0 , could not solve

dsolve(2*(x^k-4*x^3)*(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3)-(k*x^(k-1)-12*x^2)*(3*diff(y(x),x)+y(x)^2)+a*x*y(x)+b = 0, y(x),'implicit')

Mathematica raw input

DSolve[b + a*x*y[x] - (-12*x^2 + k*x^(-1 + k))*(y[x]^2 + 3*y'[x]) + 2*(-4*x^3 + x^k)*(-y[x]^3 + y[x]*y'[x] + y''[x]) == 0,y[x],x]

Mathematica raw output

DSolve[b + a*x*y[x] - (-12*x^2 + k*x^(-1 + k))*(y[x]^2 + 3*Derivative[1][y][x]) 
+ 2*(-4*x^3 + x^k)*(-y[x]^3 + y[x]*Derivative[1][y][x] + Derivative[2][y][x]) ==
 0, y[x], x]

Maple raw input

dsolve(2*(x^k-4*x^3)*(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3)-(k*x^(k-1)-12*x^2)*(3*diff(y(x),x)+y(x)^2)+a*x*y(x)+b = 0, y(x),'implicit')

Maple raw output

dsolve(2*(x^k-4*x^3)*(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3)-(k*x^(k-1)-
12*x^2)*(3*diff(y(x),x)+y(x)^2)+a*x*y(x)+b = 0, y(x),'implicit')