Internal problem ID [9270]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1691.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
Solve \begin {gather*} \boxed {\left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.055 (sec). Leaf size: 254
dsolve((a*x^2+b*x+c)^(3/2)*diff(diff(y(x),x),x)-F(y(x)/(a*x^2+b*x+c)^(1/2))=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (4 \textit {\_Z} a c -\textit {\_Z} \,b^{2}-4 F \left (\frac {\textit {\_Z}}{\sqrt {a \,x^{2}+b x +c}}\right ) \sqrt {a \,x^{2}+b x +c}\right ) \\ y \relax (x ) = \RootOf \left (-2 \left (\int _{}^{\textit {\_Z}}\frac {a}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 c a -b^{2}}+c_{2} \sqrt {4 c a -b^{2}}-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 c a -b^{2}}}\right )\right ) \sqrt {a \,x^{2}+b x +c} \\ y \relax (x ) = \RootOf \left (2 \left (\int _{}^{\textit {\_Z}}\frac {a}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 c a -b^{2}}+c_{2} \sqrt {4 c a -b^{2}}-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 c a -b^{2}}}\right )\right ) \sqrt {a \,x^{2}+b x +c} \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.465 (sec). Leaf size: 251
DSolve[-f[y[x]/Sqrt[c + b*x + a*x^2]] + (c + b*x + a*x^2)^(3/2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [2 a \text {ArcTan}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )+2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[3]^2+8 \int _1^{K[3]}f(K[2])dK[2]}}dK[3]=c_2 \sqrt {4 a c-b^2},y(x)\right ] \\ \text {Solve}\left [2 a \text {ArcTan}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )-2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[5]^2+8 \int _1^{K[5]}f(K[4])dK[4]}}dK[5]=c_2 \sqrt {4 a c-b^2},y(x)\right ] \\ \end{align*}