ODE
\[ y''(x) \left (a+2 b x+c x^2+y(x)^2\right )^2+A y(x)=0 \] ODE Classification
[NONE]
Book solution method
TO DO
Mathematica ✗
cpu = 44.8224 (sec), leaf count = 0 , could not solve
DSolve[A*y[x] + (a + 2*b*x + c*x^2 + y[x]^2)^2*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 1.051 (sec), leaf count = 334
\[ \left \{ {1 \left ( c\arctan \left ( {(cx+b){\frac {1}{\sqrt {ac-{b}^{2}}}}} \right ) +\int ^{{y \left ( x \right ) {\frac {1}{\sqrt {c{x}^{2}+2\,bx+a}}}}}\!{\frac {c}{ \left ( -ac+{b}^{2} \right ) {{\it \_f}}^{4}+ \left ( {\it \_C1}\,{c}^{2}-ac+{b}^{2} \right ) {{\it \_f}}^{2}+{\it \_C1}\,{c}^{2}+A}\sqrt { \left ( {{\it \_f}}^{2}+1 \right ) \left ( {\it \_C1}\, \left ( {{\it \_f}}^{2}+1 \right ) {c}^{2}-a{{\it \_f}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) c+{{\it \_f}}^{4}{b}^{2}+{{\it \_f}}^{2}{b}^{2}+A \right ) }}{d{\it \_f}}\sqrt {ac-{b}^{2}}-{\it \_C2}\,\sqrt {ac-{b}^{2}} \right ) {\frac {1}{\sqrt {ac-{b}^{2}}}}}=0,-{1 \left ( \int ^{{y \left ( x \right ) {\frac {1}{\sqrt {c{x}^{2}+2\,bx+a}}}}}\!{\frac {c}{ \left ( -ac+{b}^{2} \right ) {{\it \_f}}^{4}+ \left ( {\it \_C1}\,{c}^{2}-ac+{b}^{2} \right ) {{\it \_f}}^{2}+{\it \_C1}\,{c}^{2}+A}\sqrt { \left ( {{\it \_f}}^{2}+1 \right ) \left ( {\it \_C1}\, \left ( {{\it \_f}}^{2}+1 \right ) {c}^{2}-a{{\it \_f}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) c+{{\it \_f}}^{4}{b}^{2}+{{\it \_f}}^{2}{b}^{2}+A \right ) }}{d{\it \_f}}\sqrt {ac-{b}^{2}}+{\it \_C2}\,\sqrt {ac-{b}^{2}}-c\arctan \left ( {(cx+b){\frac {1}{\sqrt {ac-{b}^{2}}}}} \right ) \right ) {\frac {1}{\sqrt {ac-{b}^{2}}}}}=0 \right \} \] Mathematica raw input
DSolve[A*y[x] + (a + 2*b*x + c*x^2 + y[x]^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[A*y[x] + (a + 2*b*x + c*x^2 + y[x]^2)^2*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve((a+2*b*x+c*x^2+y(x)^2)^2*diff(diff(y(x),x),x)+A*y(x) = 0, y(x),'implicit')
Maple raw output
-(Intat(((_f^2+1)*(_C1*(_f^2+1)*c^2-a*_f^2*(_f^2+1)*c+_f^4*b^2+_f^2*b^2+A))^(1/2)*c/((-a*c+b^2)*_f^4+(_C1*c^2-a*c+b^2)*_f^2+_C1*c^2+A),_f = y(x)/(c*x^2+2*b*x+a)^(1/2))*(a*c-b^2)^(1/2)+_C2*(a*c-b^2)^(1/2)-c*arctan((c*x+b)/(a*c-b^2)^(1/2)))/(a*c-b^2)^(1/2) = 0, (c*arctan((c*x+b)/(a*c-b^2)^(1/2))+Intat(((_f^2+1)*(_C1*(_f^2+1)*c^2-a*_f^2*(_f^2+1)*c+_f^4*b^2+_f^2*b^2+A))^(1/2)*c/((-a*c+b^2)*_f^4+(_C1*c^2-a*c+b^2)*_f^2+_C1*c^2+A),_f = y(x)/(c*x^2+2*b*x+a)^(1/2))*(a*c-b^2)^(1/2)-_C2*(a*c-b^2)^(1/2))/(a*c-b^2)^(1/2) = 0