Internal problem ID [9388]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1809.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
Solve \begin {gather*} \boxed {\left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.073 (sec). Leaf size: 382
dsolve((c+2*b*x+a*x^2+y(x)^2)^2*diff(diff(y(x),x),x)+d*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {-\textit {\_f}^{6} a c +\textit {\_f}^{6} b^{2}+\textit {\_f}^{4} a^{2} c_{1}-2 \textit {\_f}^{4} a c +2 \textit {\_f}^{4} b^{2}+2 \textit {\_f}^{2} a^{2} c_{1}-c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+\textit {\_f}^{2} d +c_{1} a^{2}+d}\, a}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1}-c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+c_{1} a^{2}+d}d \textit {\_f} \right ) \sqrt {c a -b^{2}}+c_{2} \sqrt {c a -b^{2}}-a \arctan \left (\frac {a x +b}{\sqrt {c a -b^{2}}}\right )\right ) \sqrt {a \,x^{2}+2 b x +c} \\ y \relax (x ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {-\textit {\_f}^{6} a c +\textit {\_f}^{6} b^{2}+\textit {\_f}^{4} a^{2} c_{1}-2 \textit {\_f}^{4} a c +2 \textit {\_f}^{4} b^{2}+2 \textit {\_f}^{2} a^{2} c_{1}-c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+\textit {\_f}^{2} d +c_{1} a^{2}+d}\, a}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1}-c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+c_{1} a^{2}+d}d \textit {\_f} \right ) \sqrt {c a -b^{2}}+c_{2} \sqrt {c a -b^{2}}-a \arctan \left (\frac {a x +b}{\sqrt {c a -b^{2}}}\right )\right ) \sqrt {a \,x^{2}+2 b x +c} \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.931 (sec). Leaf size: 260
DSolve[d*y[x] + (c + 2*b*x + a*x^2 + y[x]^2)^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [a \text {ArcTan}\left (\frac {a x+b}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (d+\left (K[2]^2+1\right ) \left (c_1 a^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 [a \text {ArcTan}\left (\frac {a x+b}{\sqrt {a c-b^2}}\right )-\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (d+\left (K[3]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]=c_2 \sqrt {a c-b^2},y(x)\right ] \\ \end{align*}