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 = 14.6708 (sec), leaf count = 549
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\text {a0} K[1]^4+a \text {a0} K[1]^3+\text {a0} K[1]^3-c_1 K[1]^3-2 \text {a0} \log (1-K[1]) K[1]^3+\frac {\text {a2} K[1]^3}{1-K[1]}+\frac {\text {a1} K[1]^3}{a-K[1]}-a \text {a0} K[1]^2-\text {a3} K[1]^2+a c_1 K[1]^2+c_1 K[1]^2+2 a \text {a0} \log (1-K[1]) K[1]^2+2 \text {a0} \log (1-K[1]) K[1]^2-\frac {a \text {a2} K[1]^2}{1-K[1]}-\frac {\text {a2} K[1]^2}{1-K[1]}-\frac {a \text {a1} K[1]^2}{a-K[1]}-\frac {\text {a1} K[1]^2}{a-K[1]}+a \text {a3} K[1]+\text {a3} K[1]-a c_1 K[1]-2 a \text {a0} \log (1-K[1]) K[1]+\frac {a \text {a2} K[1]}{1-K[1]}+\frac {a \text {a1} K[1]}{a-K[1]}-a \text {a3}}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\text {a0} K[2]^4+a \text {a0} K[2]^3+\text {a0} K[2]^3-c_1 K[2]^3-2 \text {a0} \log (1-K[2]) K[2]^3+\frac {\text {a2} K[2]^3}{1-K[2]}+\frac {\text {a1} K[2]^3}{a-K[2]}-a \text {a0} K[2]^2-\text {a3} K[2]^2+a c_1 K[2]^2+c_1 K[2]^2+2 a \text {a0} \log (1-K[2]) K[2]^2+2 \text {a0} \log (1-K[2]) K[2]^2-\frac {a \text {a2} K[2]^2}{1-K[2]}-\frac {\text {a2} K[2]^2}{1-K[2]}-\frac {a \text {a1} K[2]^2}{a-K[2]}-\frac {\text {a1} K[2]^2}{a-K[2]}+a \text {a3} K[2]+\text {a3} K[2]-a c_1 K[2]-2 a \text {a0} \log (1-K[2]) K[2]+\frac {a \text {a2} K[2]}{1-K[2]}+\frac {a \text {a1} K[2]}{a-K[2]}-a \text {a3}}}dK[2]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.768 (sec), leaf count = 281
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-2 \ln \left (\textit {\_a} -1\right ) \textit {\_a}^{3} \mathit {a0} +2 \ln \left (\textit {\_a} -1\right ) \textit {\_a}^{2} a \mathit {a0} -\textit {\_a}^{4} \mathit {a0} +\textit {\_a}^{3} a \mathit {a0} +2 \ln \left (\textit {\_a} -1\right ) \textit {\_a}^{2} \mathit {a0} -2 \ln \left (\textit {\_a} -1\right ) \textit {\_a} a \mathit {a0} +\textit {\_a}^{3} \textit {\_C1} -\textit {\_C1} \,\textit {\_a}^{2} a +\textit {\_a}^{3} \mathit {a0} -\textit {\_a}^{2} \mathit {a0} a -\textit {\_a}^{2} \textit {\_C1} +\textit {\_C1} \textit {\_a} a -\mathit {a1} \,\textit {\_a}^{2}-\mathit {a2} \,\textit {\_a}^{2}-\mathit {a3} \,\textit {\_a}^{2}+\mathit {a2} \textit {\_a} a +\textit {\_a} a \mathit {a3} +\mathit {a1} \textit {\_a} +\mathit {a3} \textit {\_a} -a \mathit {a3}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {-2 \ln \left (\textit {\_a} -1\right ) \textit {\_a}^{3} \mathit {a0} +2 \ln \left (\textit {\_a} -1\right ) \textit {\_a}^{2} a \mathit {a0} -\textit {\_a}^{4} \mathit {a0} +\textit {\_a}^{3} a \mathit {a0} +2 \ln \left (\textit {\_a} -1\right ) \textit {\_a}^{2} \mathit {a0} -2 \ln \left (\textit {\_a} -1\right ) \textit {\_a} a \mathit {a0} +\textit {\_a}^{3} \textit {\_C1} -\textit {\_C1} \,\textit {\_a}^{2} a +\textit {\_a}^{3} \mathit {a0} -\textit {\_a}^{2} \mathit {a0} a -\textit {\_a}^{2} \textit {\_C1} +\textit {\_C1} \textit {\_a} a -\mathit {a1} \,\textit {\_a}^{2}-\mathit {a2} \,\textit {\_a}^{2}-\mathit {a3} \,\textit {\_a}^{2}+\mathit {a2} \textit {\_a} a +\textit {\_a} a \mathit {a3} +\mathit {a1} \textit {\_a} +\mathit {a3} \textit {\_a} -a \mathit {a3}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] 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
{{y[x] -> InverseFunction[Inactive[Integrate][-(1/Sqrt[-(a*a3) + a3*K[1] + a*a3*K[1] - a*C[1]*K[1] + (a*a2*K[1])/(1 - K[1]) + (a*a1*K[1])/(a - K[1]) - a*a0*K[1]^2 - a3*K[1]^2 + C[1]*K[1]^2 + a*C[1]*K[1]^2 - (a2*K[1]^2)/(1 - K[1]) - (a*a2*K[1]^2)/(1 - K[1]) - (a1*K[1]^2)/(a - K[1]) - (a*a1*K[1]^2)/(a - K[1]) + a0*K[1]^3 + a*a0*K[1]^3 - C[1]*K[1]^3 + (a2*K[1]^3)/(1 - K[1]) + (a1*K[1]^3)/(a - K[1]) - a0*K[1]^4 - 2*a*a0*K[1]*Log[1 - K[1]] + 2*a0*K[1]^2*Log[1 - K[1]] + 2*a*a0*K[1]^2*Log[1 - K[1]] - 2*a0*K[1]^3*Log[1 - K[1]]]), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][1/Sqrt[-(a*a3) + a3*K[2] + a*a3*K[2] - a*C[1]*K[2] + (a*a2*K[2])/(1 - K[2]) + (a*a1*K[2])/(a - K[2]) - a*a0*K[2]^2 - a3*K[2]^2 + C[1]*K[2]^2 + a*C[1]*K[2]^2 - (a2*K[2]^2)/(1 - K[2]) - (a*a2*K[2]^2)/(1 - K[2]) - (a1*K[2]^2)/(a - K[2]) - (a*a1*K[2]^2)/(a - K[2]) + a0*K[2]^3 + a*a0*K[2]^3 - C[1]*K[2]^3 + (a2*K[2]^3)/(1 - K[2]) + (a1*K[2]^3)/(a - K[2]) - a0*K[2]^4 - 2*a*a0*K[2]*Log[1 - K[2]] + 2*a0*K[2]^2*Log[1 - K[2]] + 2*a*a0*K[2]^2*Log[1 - K[2]] - 2*a0*K[2]^3*Log[1 - K[2]]], {K[2], 1, #1}] & ][x + C[2]]}}
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))
Maple raw output
[Intat(1/(-2*ln(_a-1)*_a^3*a0+2*ln(_a-1)*_a^2*a*a0-_a^4*a0+_a^3*a*a0+2*ln(_a-1)*_a^2*a0-2*ln(_a-1)*_a*a*a0+_a^3*_C1-_C1*_a^2*a+_a^3*a0-_a^2*a0*a-_a^2*_C1+_C1*_a*a-a1*_a^2-a2*_a^2-a3*_a^2+a2*_a*a+_a*a*a3+a1*_a+a3*_a-a*a3)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(-2*ln(_a-1)*_a^3*a0+2*ln(_a-1)*_a^2*a*a0-_a^4*a0+_a^3*a*a0+2*ln(_a-1)*_a^2*a0-2*ln(_a-1)*_a*a*a0+_a^3*_C1-_C1*_a^2*a+_a^3*a0-_a^2*a0*a-_a^2*_C1+_C1*_a*a-a1*_a^2-a2*_a^2-a3*_a^2+a2*_a*a+_a*a*a3+a1*_a+a3*_a-a*a3)^(1/2),_a = y(x))-x-_C2 = 0]