61.2.35 problem Problem 15

Internal problem ID [15273]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 15
Date solved : Friday, October 03, 2025 at 07:30:09 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {k x}{y^{4}}&=0 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 97

ode:=diff(diff(y(x),x),x)+k*x/y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (15 \sqrt {3}\, \int _{}^{\textit {\_Z}}\frac {\sqrt {-c_1 \,\textit {\_f}^{4}+50 \textit {\_f} k}\, \textit {\_f}}{c_1 \,\textit {\_f}^{3}-50 k}d \textit {\_f} x -5 c_2 x -3\right ) x \\ y &= \operatorname {RootOf}\left (15 \sqrt {3}\, \int _{}^{\textit {\_Z}}\frac {\sqrt {-c_1 \,\textit {\_f}^{4}+50 \textit {\_f} k}\, \textit {\_f}}{c_1 \,\textit {\_f}^{3}-50 k}d \textit {\_f} x +5 c_2 x +3\right ) x \\ \end{align*}

✗ Mathematica

ode=D[y[x],{x,2}]+k*x/(y[x]^4)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(k*x/y(x)**4 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve k*x/y(x)**4 + Derivative(y(x), (x, 2))