4.41.33 $2(1-y(x))y(x)(a-y(x))y^{″}(x)+(-(1-y(x))(a-y(x))+y(x)(a-y(x))+(1-y(x))y(x))y^{\prime }(x{)}^2=\text{a0}y(x{)}^2(1-y(x{)}^2)(a-y(x){)}^2+\text{a2}y(x{)}^2(a-y(x){)}^2+\text{a3}(1-y(x){)}^2(a-y(x){)}^2+\text{a1}(1-y(x){)}^{2}y(x{)}^2$

ODE
$$2(1-y(x))y(x)(a-y(x))y^{″}(x)+(-(1-y(x))(a-y(x))+y(x)(a-y(x))+(1-y(x))y(x))y^{\prime }(x{)}^2=\text{a0}y(x{)}^2(1-y(x{)}^2)(a-y(x){)}^2+\text{a2}y(x{)}^2(a-y(x){)}^2+\text{a3}(1-y(x){)}^2(a-y(x){)}^2+\text{a1}(1-y(x){)}^{2}y(x{)}^2$$ ODE Classification

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✗
cpu = 111.201 (sec), leaf count = 0 , could not solve

DSolve[(-((1 - y[x])*(a - y[x])) + (1 - y[x])*y[x] + (a - y[x])*y[x])*Derivative[1][y][x]^2 + 2*(1 - y[x])*(a - y[x])*y[x]*Derivative[2][y][x] == a3*(1 - y[x])^2*(a - y[x])^2 + a1*(1 - y[x])^2*y[x]^2 + a2*(a - y[x])^2*y[x]^2 + a0*(a - y[x])^2*y[x]^2*(1 - y[x]^2), y[x], x]

Maple ✓
cpu = 0.661 (sec), leaf count = 232

$$\{{\int }^{y\left(x\right)}\frac{1}{\sqrt{-2\ln (-1+\mathit{\_}\mathit{a}){\mathit{\_}\mathit{a}}^3\mathit{a}\mathit{0}+2\ln (-1+\mathit{\_}\mathit{a}){\mathit{\_}\mathit{a}}^{2}a\mathit{a}\mathit{0}-{\mathit{\_}\mathit{a}}^4\mathit{a}\mathit{0}+{\mathit{\_}\mathit{a}}^{3}a\mathit{a}\mathit{0}+2\ln (-1+\mathit{\_}\mathit{a}){\mathit{\_}\mathit{a}}^2\mathit{a}\mathit{0}-2\ln (-1+\mathit{\_}\mathit{a})\mathit{\_}\mathit{a}a\mathit{a}\mathit{0}+{\mathit{\_}\mathit{a}}^3\mathit{\_}\mathit{C}\mathit{1}-\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{a}}^{2}a+{\mathit{\_}\mathit{a}}^3\mathit{a}\mathit{0}-{\mathit{\_}\mathit{a}}^{2}a\mathit{a}\mathit{0}-\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{a}a-\mathit{a}\mathit{1}{\mathit{\_}\mathit{a}}^2-\mathit{a}\mathit{2}{\mathit{\_}\mathit{a}}^2-{\mathit{\_}\mathit{a}}^2\mathit{a}\mathit{3}+\mathit{a}\mathit{2}\mathit{\_}\mathit{a}a+\mathit{\_}\mathit{a}a\mathit{a}\mathit{3}+\mathit{\_}\mathit{a}\mathit{a}\mathit{1}+\mathit{\_}\mathit{a}\mathit{a}\mathit{3}-a\mathit{a}\mathit{3}}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-\frac{1}{\sqrt{2\mathit{\_}\mathit{a}\mathit{a}\mathit{0}(-1+\mathit{\_}\mathit{a})(a-\mathit{\_}\mathit{a})\ln (-1+\mathit{\_}\mathit{a})-{\mathit{\_}\mathit{a}}^4\mathit{a}\mathit{0}+(a\mathit{a}\mathit{0}+\mathit{\_}\mathit{C}\mathit{1}+\mathit{a}\mathit{0}){\mathit{\_}\mathit{a}}^3+((-\mathit{a}\mathit{0}-\mathit{\_}\mathit{C}\mathit{1})a-\mathit{a}\mathit{1}-\mathit{a}\mathit{2}-\mathit{a}\mathit{3}-\mathit{\_}\mathit{C}\mathit{1}){\mathit{\_}\mathit{a}}^2+((\mathit{a}\mathit{2}+\mathit{a}\mathit{3}+\mathit{\_}\mathit{C}\mathit{1})a+\mathit{a}\mathit{1}+\mathit{a}\mathit{3})\mathit{\_}\mathit{a}-a\mathit{a}\mathit{3}}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[(-((1 - y[x])*(a - y[x])) + (1 - y[x])*y[x] + (a - y[x])*y[x])*y'[x]^2 + 2*(1 - y[x])*(a - y[x])*y[x]*y''[x] == a3*(1 - y[x])^2*(a - y[x])^2 + a1*(1 - y[x])^2*y[x]^2 + a2*(a - y[x])^2*y[x]^2 + a0*(a - y[x])^2*y[x]^2*(1 - y[x]^2),y[x],x]

Mathematica raw output

DSolve[(-((1 - y[x])*(a - y[x])) + (1 - y[x])*y[x] + (a - y[x])*y[x])*Derivative
[1][y][x]^2 + 2*(1 - y[x])*(a - y[x])*y[x]*Derivative[2][y][x] == a3*(1 - y[x])^
2*(a - y[x])^2 + a1*(1 - y[x])^2*y[x]^2 + a2*(a - y[x])^2*y[x]^2 + a0*(a - y[x])
^2*y[x]^2*(1 - y[x]^2), y[x], x]

Maple raw input

dsolve(2*y(x)*(1-y(x))*(a-y(x))*diff(diff(y(x),x),x)+(y(x)*(1-y(x))+y(x)*(a-y(x))-(1-y(x))*(a-y(x)))*diff(y(x),x)^2 = a0*y(x)^2*(1-y(x)^2)*(a-y(x))^2+a1*y(x)^2*(1-y(x))^2+a2*y(x)^2*(a-y(x))^2+a3*(1-y(x))^2*(a-y(x))^2, y(x),'implicit')

Maple raw output

Intat(1/(-2*ln(-1+_a)*_a^3*a0+2*ln(-1+_a)*_a^2*a*a0-_a^4*a0+_a^3*a*a0+2*ln(-1+_a
)*_a^2*a0-2*ln(-1+_a)*_a*a*a0+_a^3*_C1-_C1*_a^2*a+_a^3*a0-_a^2*a*a0-_C1*_a^2+_C1
*_a*a-a1*_a^2-a2*_a^2-_a^2*a3+a2*_a*a+_a*a*a3+_a*a1+_a*a3-a*a3)^(1/2),_a = y(x))
-x-_C2 = 0, Intat(-1/(2*_a*a0*(-1+_a)*(a-_a)*ln(-1+_a)-_a^4*a0+(a*a0+_C1+a0)*_a^
3+((-a0-_C1)*a-a1-a2-a3-_C1)*_a^2+((a2+a3+_C1)*a+a1+a3)*_a-a*a3)^(1/2),_a = y(x)
)-x-_C2 = 0