4.41.35 $2(1-x)x(1-y(x))(x-y(x))y(x)y^{″}(x)=2(1-y(x))(x^2-2xy(x)+y(x))y(x)y^{\prime }(x)+(1-x)x(3y(x{)}^2-2xy(x)-2y(x)+x)y^{\prime }(x{)}^2-(1-y(x{)}^2)y(x{)}^2$

ODE
$$2(1-x)x(1-y(x))(x-y(x))y(x)y^{″}(x)=2(1-y(x))(x^2-2xy(x)+y(x))y(x)y^{\prime }(x)+(1-x)x(3y(x{)}^2-2xy(x)-2y(x)+x)y^{\prime }(x{)}^2-(1-y(x{)}^2)y(x{)}^2$$ ODE Classification

[NONE]

Book solution method
TO DO

Mathematica ✗
cpu = 6.95114 (sec), leaf count = 0 , could not solve

DSolve[2*(1 - x)*x*(1 - y[x])*(x - y[x])*y[x]*Derivative[2][y][x] == -(y[x]^2*(1 - y[x]^2)) + 2*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*Derivative[1][y][x] + (1 - x)*x*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*Derivative[1][y][x]^2, y[x], x]

Maple ✓
cpu = 2.211 (sec), leaf count = 5

$$\{y\left(x\right)=1\}$$ Mathematica raw input

DSolve[2*(1 - x)*x*(1 - y[x])*(x - y[x])*y[x]*y''[x] == -(y[x]^2*(1 - y[x]^2)) + 2*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*y'[x] + (1 - x)*x*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*y'[x]^2,y[x],x]

Mathematica raw output

DSolve[2*(1 - x)*x*(1 - y[x])*(x - y[x])*y[x]*Derivative[2][y][x] == -(y[x]^2*(1
 - y[x]^2)) + 2*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*Derivative[1][y][x] + (1
 - x)*x*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*Derivative[1][y][x]^2, y[x], x]

Maple raw input

dsolve(2*x*y(x)*(1-x)*(1-y(x))*(x-y(x))*diff(diff(y(x),x),x) = x*(1-x)*(x-2*x*y(x)-2*y(x)+3*y(x)^2)*diff(y(x),x)^2+2*y(x)*(1-y(x))*(x^2+y(x)-2*x*y(x))*diff(y(x),x)-y(x)^2*(1-y(x)^2), y(x),'implicit')

Maple raw output

y(x) = 1