ODE
\[ y(x) y''(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3+\text {a4} y(x)^4+y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 3.33586 (sec), leaf count = 125
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {\text {a4} K[1]^4+2 \text {a3} K[1]^3+c_1 K[1]^2+2 \text {a2} \log (K[1]) K[1]^2-2 \text {a1} K[1]-\text {a0}}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {\text {a4} K[2]^4+2 \text {a3} K[2]^3+c_1 K[2]^2+2 \text {a2} \log (K[2]) K[2]^2-2 \text {a1} K[2]-\text {a0}}}dK[2]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.708 (sec), leaf count = 99
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\mathit {a4} \,\textit {\_a}^{4}+2 \ln \left (\textit {\_a} \right ) \textit {\_a}^{2} \mathit {a2} +2 \mathit {a3} \,\textit {\_a}^{3}+\textit {\_a}^{2} \textit {\_C1} -2 \mathit {a1} \textit {\_a} -\mathit {a0}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {\mathit {a4} \,\textit {\_a}^{4}+2 \ln \left (\textit {\_a} \right ) \textit {\_a}^{2} \mathit {a2} +2 \mathit {a3} \,\textit {\_a}^{3}+\textit {\_a}^{2} \textit {\_C1} -2 \mathit {a1} \textit {\_a} -\mathit {a0}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == a0 + a1*y[x] + a2*y[x]^2 + a3*y[x]^3 + a4*y[x]^4 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][-(1/Sqrt[-a0 - 2*a1*K[1] + C[1]*K[1]^2 + 2*a3*K[1]^3 + a4*K[1]^4 + 2*a2*K[1]^2*Log[K[1]]]), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][1/Sqrt[-a0 - 2*a1*K[2] + C[1]*K[2]^2 + 2*a3*K[2]^3 + a4*K[2]^4 + 2*a2*K[2]^2*Log[K[2]]], {K[2], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+a0+a1*y(x)+a2*y(x)^2+a3*y(x)^3+a4*y(x)^4, y(x))
Maple raw output
[Intat(1/(a4*_a^4+2*ln(_a)*_a^2*a2+2*a3*_a^3+_a^2*_C1-2*a1*_a-a0)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(a4*_a^4+2*ln(_a)*_a^2*a2+2*a3*_a^3+_a^2*_C1-2*a1*_a-a0)^(1/2),_a = y(x))-x-_C2 = 0]