54.7.20 problem 1610 (6.20)

Internal problem ID [12869]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1610 (6.20)
Date solved : Friday, October 03, 2025 at 03:47:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}} \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 92

ode:=diff(diff(y(x),x),x) = 1/x^(3/2)*f(y(x)/x^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (\textit {\_Z} \,x^{{3}/{2}}+4 f \left (\frac {\textit {\_Z}}{\sqrt {x}}\right ) x^{2}\right ) \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_1 +8 \int f \left (\textit {\_g} \right )d \textit {\_g} +\textit {\_g}^{2}}}d \textit {\_g} +2 c_2 \right ) \sqrt {x} \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-2 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_1 +8 \int f \left (\textit {\_g} \right )d \textit {\_g} +\textit {\_g}^{2}}}d \textit {\_g} +2 c_2 \right ) \sqrt {x} \\ \end{align*}

✓ Mathematica. Time used: 2.772 (sec). Leaf size: 754

ode=-(f[y[x]*x^(-1/2)]*x^(-3/2)) + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},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}}}f(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]}}}f(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]}}}f(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]}}}f(K[2])dK[2]-\frac {4 f\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]}}}f(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]}}}f(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]}}}f(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}}}f(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]}}}f(K[2])dK[2]-\frac {4 f\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]}}}f(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]}}}f(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]}}}f(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]}}}f(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]}}}f(K[2])dK[2]}{K[6]}}}\right )dK[6]=c_2,y(x)\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq(Derivative(y(x), (x, 2)) - f(y(x)/sqrt(x))/x**(3/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve Derivative(y(x), (x, 2)) - f(y(x)/sqrt(x))/x**(3/2)