ODE
\[ y(x) y''(x)=a y'(x)^2+\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3+\text {a3} y(x)^2+\text {a4} y(x)^4 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.43304 (sec), leaf count = 646
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^5-20 a^4+35 a^3-25 a^2+6 a}}{\sqrt {4 a^5 c_1 K[1]^{2 a}-20 a^4 c_1 K[1]^{2 a}+35 a^3 c_1 K[1]^{2 a}-25 a^2 c_1 K[1]^{2 a}+6 a c_1 K[1]^{2 a}-4 a^4 \text {a4} K[1]^4+12 a^3 \text {a4} K[1]^4-11 a^2 \text {a4} K[1]^4+3 a \text {a4} K[1]^4-4 a^4 \text {a3} K[1]^3+14 a^3 \text {a3} K[1]^3-14 a^2 \text {a3} K[1]^3+4 a \text {a3} K[1]^3-4 a^4 \text {a2} K[1]^2+16 a^3 \text {a2} K[1]^2-19 a^2 \text {a2} K[1]^2+6 a \text {a2} K[1]^2-4 a^4 \text {a3} K[1]^2+16 a^3 \text {a3} K[1]^2-19 a^2 \text {a3} K[1]^2+6 a \text {a3} K[1]^2-4 a^4 \text {a1} K[1]+18 a^3 \text {a1} K[1]-26 a^2 \text {a1} K[1]+12 a \text {a1} K[1]-4 a^4 \text {a0}+20 a^3 \text {a0}-35 a^2 \text {a0}+25 a \text {a0}-6 \text {a0}}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^5-20 a^4+35 a^3-25 a^2+6 a}}{\sqrt {4 a^5 c_1 K[2]^{2 a}-20 a^4 c_1 K[2]^{2 a}+35 a^3 c_1 K[2]^{2 a}-25 a^2 c_1 K[2]^{2 a}+6 a c_1 K[2]^{2 a}-4 a^4 \text {a4} K[2]^4+12 a^3 \text {a4} K[2]^4-11 a^2 \text {a4} K[2]^4+3 a \text {a4} K[2]^4-4 a^4 \text {a3} K[2]^3+14 a^3 \text {a3} K[2]^3-14 a^2 \text {a3} K[2]^3+4 a \text {a3} K[2]^3-4 a^4 \text {a2} K[2]^2+16 a^3 \text {a2} K[2]^2-19 a^2 \text {a2} K[2]^2+6 a \text {a2} K[2]^2-4 a^4 \text {a3} K[2]^2+16 a^3 \text {a3} K[2]^2-19 a^2 \text {a3} K[2]^2+6 a \text {a3} K[2]^2-4 a^4 \text {a1} K[2]+18 a^3 \text {a1} K[2]-26 a^2 \text {a1} K[2]+12 a \text {a1} K[2]-4 a^4 \text {a0}+20 a^3 \text {a0}-35 a^2 \text {a0}+25 a \text {a0}-6 \text {a0}}}dK[2]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 1.167 (sec), leaf count = 604
\[\left [\int _{}^{y \left (x \right )}\frac {a \left (4 a^{4}-20 a^{3}+35 a^{2}-25 a +6\right )}{\sqrt {a \left (4 a^{4}-20 a^{3}+35 a^{2}-25 a +6\right ) \left (-6 \mathit {a0} -19 \textit {\_a}^{2} a^{2} \mathit {a2} -19 \textit {\_a}^{2} a^{2} \mathit {a3} +18 \textit {\_a} \,a^{3} \mathit {a1} +6 \textit {\_a}^{2} a \mathit {a2} +6 \textit {\_a}^{2} a \mathit {a3} -26 \textit {\_a} \,a^{2} \mathit {a1} +12 \textit {\_a} a \mathit {a1} -4 a^{4} \mathit {a0} +20 a^{3} \mathit {a0} +25 \mathit {a0} a -4 \textit {\_a}^{4} a^{4} \mathit {a4} +12 \textit {\_a}^{4} a^{3} \mathit {a4} -4 \textit {\_a}^{3} a^{4} \mathit {a3} -11 \textit {\_a}^{4} a^{2} \mathit {a4} +14 \textit {\_a}^{3} a^{3} \mathit {a3} -4 \textit {\_a}^{2} a^{4} \mathit {a2} -4 \textit {\_a}^{2} a^{4} \mathit {a3} +3 \textit {\_a}^{4} a \mathit {a4} -14 \textit {\_a}^{3} a^{2} \mathit {a3} +16 \textit {\_a}^{2} a^{3} \mathit {a2} +16 \textit {\_a}^{2} a^{3} \mathit {a3} -4 \textit {\_a} \,a^{4} \mathit {a1} +4 \textit {\_a}^{3} a \mathit {a3} -20 \textit {\_a}^{2 a} \textit {\_C1} \,a^{4}-35 a^{2} \mathit {a0} +35 \textit {\_a}^{2 a} \textit {\_C1} \,a^{3}-25 \textit {\_a}^{2 a} \textit {\_C1} \,a^{2}+6 \textit {\_C1} \,\textit {\_a}^{2 a} a +4 \textit {\_a}^{2 a} \textit {\_C1} \,a^{5}\right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {a \left (4 a^{4}-20 a^{3}+35 a^{2}-25 a +6\right )}{\sqrt {a \left (4 a^{4}-20 a^{3}+35 a^{2}-25 a +6\right ) \left (-6 \mathit {a0} -19 \textit {\_a}^{2} a^{2} \mathit {a2} -19 \textit {\_a}^{2} a^{2} \mathit {a3} +18 \textit {\_a} \,a^{3} \mathit {a1} +6 \textit {\_a}^{2} a \mathit {a2} +6 \textit {\_a}^{2} a \mathit {a3} -26 \textit {\_a} \,a^{2} \mathit {a1} +12 \textit {\_a} a \mathit {a1} -4 a^{4} \mathit {a0} +20 a^{3} \mathit {a0} +25 \mathit {a0} a -4 \textit {\_a}^{4} a^{4} \mathit {a4} +12 \textit {\_a}^{4} a^{3} \mathit {a4} -4 \textit {\_a}^{3} a^{4} \mathit {a3} -11 \textit {\_a}^{4} a^{2} \mathit {a4} +14 \textit {\_a}^{3} a^{3} \mathit {a3} -4 \textit {\_a}^{2} a^{4} \mathit {a2} -4 \textit {\_a}^{2} a^{4} \mathit {a3} +3 \textit {\_a}^{4} a \mathit {a4} -14 \textit {\_a}^{3} a^{2} \mathit {a3} +16 \textit {\_a}^{2} a^{3} \mathit {a2} +16 \textit {\_a}^{2} a^{3} \mathit {a3} -4 \textit {\_a} \,a^{4} \mathit {a1} +4 \textit {\_a}^{3} a \mathit {a3} -20 \textit {\_a}^{2 a} \textit {\_C1} \,a^{4}-35 a^{2} \mathit {a0} +35 \textit {\_a}^{2 a} \textit {\_C1} \,a^{3}-25 \textit {\_a}^{2 a} \textit {\_C1} \,a^{2}+6 \textit {\_C1} \,\textit {\_a}^{2 a} a +4 \textit {\_a}^{2 a} \textit {\_C1} \,a^{5}\right )}}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]^2 + a3*y[x]^3 + a4*y[x]^4 + a*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][-(Sqrt[6*a - 25*a^2 + 35*a^3 - 20*a^4 + 4*a^5]/Sqrt[-6*a0 + 25*a*a0 - 35*a^2*a0 + 20*a^3*a0 - 4*a^4*a0 + 12*a*a1*K[1] - 26*a^2*a1*K[1] + 18*a^3*a1*K[1] - 4*a^4*a1*K[1] + 6*a*a2*K[1]^2 - 19*a^2*a2*K[1]^2 + 16*a^3*a2*K[1]^2 - 4*a^4*a2*K[1]^2 + 6*a*a3*K[1]^2 - 19*a^2*a3*K[1]^2 + 16*a^3*a3*K[1]^2 - 4*a^4*a3*K[1]^2 + 4*a*a3*K[1]^3 - 14*a^2*a3*K[1]^3 + 14*a^3*a3*K[1]^3 - 4*a^4*a3*K[1]^3 + 3*a*a4*K[1]^4 - 11*a^2*a4*K[1]^4 + 12*a^3*a4*K[1]^4 - 4*a^4*a4*K[1]^4 + 6*a*C[1]*K[1]^(2*a) - 25*a^2*C[1]*K[1]^(2*a) + 35*a^3*C[1]*K[1]^(2*a) - 20*a^4*C[1]*K[1]^(2*a) + 4*a^5*C[1]*K[1]^(2*a)]), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][Sqrt[6*a - 25*a^2 + 35*a^3 - 20*a^4 + 4*a^5]/Sqrt[-6*a0 + 25*a*a0 - 35*a^2*a0 + 20*a^3*a0 - 4*a^4*a0 + 12*a*a1*K[2] - 26*a^2*a1*K[2] + 18*a^3*a1*K[2] - 4*a^4*a1*K[2] + 6*a*a2*K[2]^2 - 19*a^2*a2*K[2]^2 + 16*a^3*a2*K[2]^2 - 4*a^4*a2*K[2]^2 + 6*a*a3*K[2]^2 - 19*a^2*a3*K[2]^2 + 16*a^3*a3*K[2]^2 - 4*a^4*a3*K[2]^2 + 4*a*a3*K[2]^3 - 14*a^2*a3*K[2]^3 + 14*a^3*a3*K[2]^3 - 4*a^4*a3*K[2]^3 + 3*a*a4*K[2]^4 - 11*a^2*a4*K[2]^4 + 12*a^3*a4*K[2]^4 - 4*a^4*a4*K[2]^4 + 6*a*C[1]*K[2]^(2*a) - 25*a^2*C[1]*K[2]^(2*a) + 35*a^3*C[1]*K[2]^(2*a) - 20*a^4*C[1]*K[2]^(2*a) + 4*a^5*C[1]*K[2]^(2*a)], {K[2], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a*diff(y(x),x)^2+a0+a1*y(x)+a2*y(x)^2+a3*y(x)^2+a3*y(x)^3+a4*y(x)^4, y(x))
Maple raw output
[Intat(1/(a*(4*a^4-20*a^3+35*a^2-25*a+6)*(-6*a0-19*_a^2*a^2*a2-19*_a^2*a^2*a3+18*_a*a^3*a1+6*_a^2*a*a2+6*_a^2*a*a3-26*_a*a^2*a1+12*_a*a*a1-4*a^4*a0+20*a^3*a0+25*a0*a-4*_a^4*a^4*a4+12*_a^4*a^3*a4-4*_a^3*a^4*a3-11*_a^4*a^2*a4+14*_a^3*a^3*a3-4*_a^2*a^4*a2-4*_a^2*a^4*a3+3*_a^4*a*a4-14*_a^3*a^2*a3+16*_a^2*a^3*a2+16*_a^2*a^3*a3-4*_a*a^4*a1+4*_a^3*a*a3-20*_a^(2*a)*_C1*a^4-35*a^2*a0+35*_a^(2*a)*_C1*a^3-25*_a^(2*a)*_C1*a^2+6*_C1*_a^(2*a)*a+4*_a^(2*a)*_C1*a^5))^(1/2)*a*(4*a^4-20*a^3+35*a^2-25*a+6),_a = y(x))-x-_C2 = 0, Intat(-1/(a*(4*a^4-20*a^3+35*a^2-25*a+6)*(-6*a0-19*_a^2*a^2*a2-19*_a^2*a^2*a3+18*_a*a^3*a1+6*_a^2*a*a2+6*_a^2*a*a3-26*_a*a^2*a1+12*_a*a*a1-4*a^4*a0+20*a^3*a0+25*a0*a-4*_a^4*a^4*a4+12*_a^4*a^3*a4-4*_a^3*a^4*a3-11*_a^4*a^2*a4+14*_a^3*a^3*a3-4*_a^2*a^4*a2-4*_a^2*a^4*a3+3*_a^4*a*a4-14*_a^3*a^2*a3+16*_a^2*a^3*a2+16*_a^2*a^3*a3-4*_a*a^4*a1+4*_a^3*a*a3-20*_a^(2*a)*_C1*a^4-35*a^2*a0+35*_a^(2*a)*_C1*a^3-25*_a^(2*a)*_C1*a^2+6*_C1*_a^(2*a)*a+4*_a^(2*a)*_C1*a^5))^(1/2)*a*(4*a^4-20*a^3+35*a^2-25*a+6),_a = y(x))-x-_C2 = 0]