Internal problem ID [9189]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1610.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {h \left (\frac {y}{\sqrt {x}}\right )}{x^{\frac {3}{2}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.03 (sec). Leaf size: 92
dsolve(diff(diff(y(x),x),x)-1/x^(3/2)*h(y(x)/x^(1/2))=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (\textit {\_Z} \,x^{\frac {3}{2}}+4 h \left (\frac {\textit {\_Z}}{\sqrt {x}}\right ) x^{2}\right ) \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_{1}+8 \left (\int h \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 c_{2}\right ) \sqrt {x} \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_{1}+8 \left (\int h \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 c_{2}\right ) \sqrt {x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.418 (sec). Leaf size: 754
DSolve[-(h[y[x]/Sqrt[x]]/x^(3/2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\int _1^{y(x)}\frac {2}{\sqrt {x} \sqrt {\frac {K[3]^2+4 x c_1+8 x \int _1^{\frac {K[3]}{\sqrt {x}}}h(K[2])dK[2]}{x}}}dK[3]-\int _1^x\left (\frac {2 \left (\frac {y(x)}{2 \sqrt {K[4]}}-\frac {\sqrt {\frac {y(x)^2}{2 K[4]}+2 c_1+4 \int _1^{\frac {y(x)}{\sqrt {K[4]}}}h(K[2])dK[2]}}{\sqrt {2}}\right )}{K[4] \sqrt {\frac {y(x)^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {y(x)}{\sqrt {K[4]}}}h(K[2])dK[2]}{K[4]}}}+\int _1^{y(x)}\left (-\frac {\frac {4 c_1+8 \int _1^{\frac {K[3]}{\sqrt {K[4]}}}h(K[2])dK[2]-\frac {4 h\left (\frac {K[3]}{\sqrt {K[4]}}\right ) K[3]}{\sqrt {K[4]}}}{K[4]}-\frac {K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}h(K[2])dK[2]}{K[4]^2}}{\sqrt {K[4]} \left (\frac {K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}h(K[2])dK[2]}{K[4]}\right ){}^{3/2}}-\frac {1}{K[4]^{3/2} \sqrt {\frac {K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}h(K[2])dK[2]}{K[4]}}}\right )dK[3]\right )dK[4]=c_2,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}-\frac {2}{\sqrt {x} \sqrt {\frac {K[5]^2+4 x c_1+8 x \int _1^{\frac {K[5]}{\sqrt {x}}}h(K[2])dK[2]}{x}}}dK[5]-\int _1^x\left (\int _1^{y(x)}\left (\frac {\frac {4 c_1+8 \int _1^{\frac {K[5]}{\sqrt {K[6]}}}h(K[2])dK[2]-\frac {4 h\left (\frac {K[5]}{\sqrt {K[6]}}\right ) K[5]}{\sqrt {K[6]}}}{K[6]}-\frac {K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}h(K[2])dK[2]}{K[6]^2}}{\sqrt {K[6]} \left (\frac {K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}h(K[2])dK[2]}{K[6]}\right ){}^{3/2}}+\frac {1}{K[6]^{3/2} \sqrt {\frac {K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}h(K[2])dK[2]}{K[6]}}}\right )dK[5]-\frac {2 \left (\frac {y(x)}{2 \sqrt {K[6]}}+\frac {\sqrt {\frac {y(x)^2}{2 K[6]}+2 c_1+4 \int _1^{\frac {y(x)}{\sqrt {K[6]}}}h(K[2])dK[2]}}{\sqrt {2}}\right )}{K[6] \sqrt {\frac {y(x)^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {y(x)}{\sqrt {K[6]}}}h(K[2])dK[2]}{K[6]}}}\right )dK[6]=c_2,y(x)\right ] \\ \end{align*}