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 = 103.826 (sec), leaf count = 0 , could not solve
DSolve[y[x]*Derivative[2][y][x] == a0 + a1*y[x] + a2*y[x]^2 + a3*y[x]^2 + a3*y[x]^3 + a4*y[x]^4 + a*Derivative[1][y][x]^2, y[x], x]
Maple ✓
cpu = 0.19 (sec), leaf count = 313
\[ \left \{ \int ^{y \left ( x \right ) }\!-{\frac {\sqrt {16}a \left ( 4\,{a}^{4}-20\,{a}^{3}+35\,{a}^{2}-25\,a+6 \right ) }{16}{\frac {1}{\sqrt { \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) a \left ( \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) a{\it \_C1}\,{{\it \_a}}^{2\,a}- \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) {\it a4}\,a{{\it \_a}}^{4}- \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-2 \right ) {\it a3}\,a{{\it \_a}}^{3}- \left ( a-{\frac {1}{2}} \right ) \left ( {\it a2}+{\it a3} \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) a{{\it \_a}}^{2}- \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) {\it a1}\,a{\it \_a}- \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) {\it a0}\, \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{\frac {\sqrt {16}a \left ( 4\,{a}^{4}-20\,{a}^{3}+35\,{a}^{2}-25\,a+6 \right ) }{16}{\frac {1}{\sqrt { \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) a \left ( \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) a{\it \_C1}\,{{\it \_a}}^{2\,a}- \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) {\it a4}\,a{{\it \_a}}^{4}- \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) \left ( a-2 \right ) {\it a3}\,a{{\it \_a}}^{3}- \left ( a-{\frac {1}{2}} \right ) \left ( {\it a2}+{\it a3} \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) a{{\it \_a}}^{2}- \left ( a-1 \right ) \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) {\it a1}\,a{\it \_a}- \left ( a-{\frac {1}{2}} \right ) \left ( a-1 \right ) {\it a0}\, \left ( a-{\frac {3}{2}} \right ) \left ( a-2 \right ) \right ) }}}}{d{\it \_a}}-x-{\it \_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
DSolve[y[x]*Derivative[2][y][x] == a0 + a1*y[x] + a2*y[x]^2 + a3*y[x]^2 + a3*y[x]^3 + a4*y[x]^4 + a*Derivative[1][y][x]^2, y[x], x]
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),'implicit')
Maple raw output
Intat(1/16*16^(1/2)/((a-1/2)*(a-1)*(a-3/2)*(a-2)*a*((a-1/2)*(a-1)*(a-3/2)*(a-2)*a*_C1*_a^(2*a)-(a-1/2)*(a-1)*(a-3/2)*a4*a*_a^4-(a-1/2)*(a-1)*(a-2)*a3*a*_a^3-(a-1/2)*(a2+a3)*(a-3/2)*(a-2)*a*_a^2-(a-1)*(a-3/2)*(a-2)*a1*a*_a-(a-1/2)*(a-1)*a0*(a-3/2)*(a-2)))^(1/2)*a*(4*a^4-20*a^3+35*a^2-25*a+6),_a = y(x))-x-_C2 = 0, Intat(-1/16*16^(1/2)/((a-1/2)*(a-1)*(a-3/2)*(a-2)*a*((a-1/2)*(a-1)*(a-3/2)*(a-2)*a*_C1*_a^(2*a)-(a-1/2)*(a-1)*(a-3/2)*a4*a*_a^4-(a-1/2)*(a-1)*(a-2)*a3*a*_a^3-(a-1/2)*(a2+a3)*(a-3/2)*(a-2)*a*_a^2-(a-1)*(a-3/2)*(a-2)*a1*a*_a-(a-1/2)*(a-1)*a0*(a-3/2)*(a-2)))^(1/2)*a*(4*a^4-20*a^3+35*a^2-25*a+6),_a = y(x))-x-_C2 = 0