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′(x)2 = a0 y(x)2 ( 1 −y(x)2) (a −y(x))2 + a2y(x)2(a −y(x))2 + a3(1 −y(x))2(a −y(x))2 + a1(1 −y(x))2y(x)2

4.41.33 \(2 (1-y(x)) y(x) (a-y(x)) y''(x)+(-(1-y(x)) (a-y(x))+y(x) (a-y(x))+(1-y(x)) y(x)) y'(x)^2=\text {a0} y(x)^2 \left (1-y(x)^2\right ) (a-y(x))^2+\text {a2} y(x)^2 (a-y(x))^2+\text {a3} (1-y(x))^2 (a-y(x))^2+\text {a1} (1-y(x))^2 y(x)^2\)

ODE
\[ 2 (1-y(x)) y(x) (a-y(x)) y''(x)+(-(1-y(x)) (a-y(x))+y(x) (a-y(x))+(1-y(x)) y(x)) y'(x)^2=\text {a0} y(x)^2 \left (1-y(x)^2\right ) (a-y(x))^2+\text {a2} y(x)^2 (a-y(x))^2+\text {a3} (1-y(x))^2 (a-y(x))^2+\text {a1} (1-y(x))^2 y(x)^2 \] 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

\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {-2\,\ln \left ( -1+{\it \_a} \right ) {{\it \_a}}^{3}{\it a0}+2\,\ln \left ( -1+{\it \_a} \right ) {{\it \_a}}^{2}a{\it a0}-{{\it \_a}}^{4}{\it a0}+{{\it \_a}}^{3}a{\it a0}+2\,\ln \left ( -1+{\it \_a} \right ) {{\it \_a}}^{2}{\it a0}-2\,\ln \left ( -1+{\it \_a} \right ) {\it \_a}\,a{\it a0}+{{\it \_a}}^{3}{\it \_C1}-{\it \_C1}\,{{\it \_a}}^{2}a+{{\it \_a}}^{3}{\it a0}-{{\it \_a}}^{2}a{\it a0}-{\it \_C1}\,{{\it \_a}}^{2}+{\it \_C1}\,{\it \_a}\,a-{\it a1}\,{{\it \_a}}^{2}-{\it a2}\,{{\it \_a}}^{2}-{{\it \_a}}^{2}{\it a3}+{\it a2}\,{\it \_a}\,a+{\it \_a}\,a{\it a3}+{\it \_a}\,{\it a1}+{\it \_a}\,{\it a3}-a{\it a3}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt {2\,{\it \_a}\,{\it a0}\, \left ( -1+{\it \_a} \right ) \left ( a-{\it \_a} \right ) \ln \left ( -1+{\it \_a} \right ) -{{\it \_a}}^{4}{\it a0}+ \left ( a{\it a0}+{\it \_C1}+{\it a0} \right ) {{\it \_a}}^{3}+ \left ( \left ( -{\it a0}-{\it \_C1} \right ) a-{\it a1}-{\it a2}-{\it a3}-{\it \_C1} \right ) {{\it \_a}}^{2}+ \left ( \left ( {\it a2}+{\it a3}+{\it \_C1} \right ) a+{\it a1}+{\it a3} \right ) {\it \_a}-a{\it a3}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] 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

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