4.41.40 \(y''(x) \left (a+2 b x+c x^2+y(x)^2\right )^2+A y(x)=0\)

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 = 24.9817 (sec), leaf count = 245

\[\left \{\text {Solve}\left [c \tan ^{-1}\left (\frac {b+c x}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {a+x (2 b+c x)}}}\frac {c \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (A+\left (K[2]^2+1\right ) \left (c_1 c^2+\left (b^2-a c\right ) K[2]^2\right )\right )}}dK[2]=c_2 \sqrt {a c-b^2},y(x)\right ],\text {Solve}\left [c \tan ^{-1}\left (\frac {b+c x}{\sqrt {a c-b^2}}\right )=\sqrt {a c-b^2} \left (\int _1^{\frac {y(x)}{\sqrt {a+x (2 b+c x)}}}\frac {c \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (A+\left (K[3]^2+1\right ) \left (c_1 c^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]+c_2\right ),y(x)\right ]\right \}\]

Maple ✓
cpu = 1.646 (sec), leaf count = 382

\[\left [y \left (x \right ) = \RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {-\textit {\_f}^{6} a c +\textit {\_f}^{6} b^{2}+\textit {\_C1} \,\textit {\_f}^{4} c^{2}-2 \textit {\_f}^{4} a c +2 \textit {\_f}^{4} b^{2}+2 \textit {\_C1} \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+A \,\textit {\_f}^{2}+\textit {\_C1} \,c^{2}+A}\, c}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_C1} \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+\textit {\_C1} \,c^{2}+A}d \textit {\_f} \right ) \sqrt {a c -b^{2}}+\textit {\_C2} \sqrt {a c -b^{2}}-\arctan \left (\frac {c x +b}{\sqrt {a c -b^{2}}}\right ) c \right ) \sqrt {c \,x^{2}+2 b x +a}, y \left (x \right ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {-\textit {\_f}^{6} a c +\textit {\_f}^{6} b^{2}+\textit {\_C1} \,\textit {\_f}^{4} c^{2}-2 \textit {\_f}^{4} a c +2 \textit {\_f}^{4} b^{2}+2 \textit {\_C1} \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+A \,\textit {\_f}^{2}+\textit {\_C1} \,c^{2}+A}\, c}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_C1} \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+\textit {\_C1} \,c^{2}+A}d \textit {\_f} \right ) \sqrt {a c -b^{2}}+\textit {\_C2} \sqrt {a c -b^{2}}-\arctan \left (\frac {c x +b}{\sqrt {a c -b^{2}}}\right ) c \right ) \sqrt {c \,x^{2}+2 b x +a}\right ]\] 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

{Solve[c*ArcTan[(b + c*x)/Sqrt[-b^2 + a*c]] + Sqrt[-b^2 + a*c]*Inactive[Integrat
e][(c*(1 + K[2]^2))/Sqrt[(1 + K[2]^2)*(A + (1 + K[2]^2)*(c^2*C[1] + (b^2 - a*c)*
K[2]^2))], {K[2], 1, y[x]/Sqrt[a + x*(2*b + c*x)]}] == Sqrt[-b^2 + a*c]*C[2], y[
x]], Solve[c*ArcTan[(b + c*x)/Sqrt[-b^2 + a*c]] == Sqrt[-b^2 + a*c]*(C[2] + Inac
tive[Integrate][(c*(1 + K[3]^2))/Sqrt[(1 + K[3]^2)*(A + (1 + K[3]^2)*(c^2*C[1] +
 (b^2 - a*c)*K[3]^2))], {K[3], 1, y[x]/Sqrt[a + x*(2*b + c*x)]}]), y[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))

Maple raw output

[y(x) = RootOf(Intat(1/(-_f^4*a*c+_f^4*b^2+_C1*_f^2*c^2-_f^2*a*c+_f^2*b^2+_C1*c^
2+A)*(-_f^6*a*c+_f^6*b^2+_C1*_f^4*c^2-2*_f^4*a*c+2*_f^4*b^2+2*_C1*_f^2*c^2-_f^2*
a*c+_f^2*b^2+A*_f^2+_C1*c^2+A)^(1/2)*c,_f = _Z)*(a*c-b^2)^(1/2)+_C2*(a*c-b^2)^(1
/2)-arctan((c*x+b)/(a*c-b^2)^(1/2))*c)*(c*x^2+2*b*x+a)^(1/2), y(x) = RootOf(-Int
at(1/(-_f^4*a*c+_f^4*b^2+_C1*_f^2*c^2-_f^2*a*c+_f^2*b^2+_C1*c^2+A)*(-_f^6*a*c+_f
^6*b^2+_C1*_f^4*c^2-2*_f^4*a*c+2*_f^4*b^2+2*_C1*_f^2*c^2-_f^2*a*c+_f^2*b^2+A*_f^
2+_C1*c^2+A)^(1/2)*c,_f = _Z)*(a*c-b^2)^(1/2)+_C2*(a*c-b^2)^(1/2)-arctan((c*x+b)
/(a*c-b^2)^(1/2))*c)*(c*x^2+2*b*x+a)^(1/2)]