54.7.83 problem 1692 (book 6.101)

Internal problem ID [12932]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1692 (book 6.101)
Date solved : Friday, October 03, 2025 at 03:50:51 AM
CAS classification : [NONE]

\begin{align*} \left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right )&=0 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 252

ode:=(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; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {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 &= \operatorname {RootOf}\left (-2 a \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_1 \,a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \int F \left (\textit {\_g} \right )d \textit {\_g}}}d \textit {\_g} \sqrt {4 a c -b^{2}}-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_2 \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\ y &= \operatorname {RootOf}\left (2 a \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_1 \,a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \int F \left (\textit {\_g} \right )d \textit {\_g}}}d \textit {\_g} \sqrt {4 a c -b^{2}}-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_2 \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\ \end{align*}

✓ Mathematica. Time used: 6.955 (sec). Leaf size: 251

ode=-f[y[x]/Sqrt[c + b*x + a*x^2]] + (c + b*x + a*x^2)^(3/2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [2 a \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 \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*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve (a*x**2 + b*x + c)**(3/2)*Derivative(y(x), (x, 2)) - F(y(x)/sqrt(a*x**2 + b*x + c))