54.7.137 problem 1753 (book 6.162)

Internal problem ID [12986]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1753 (book 6.162)
Date solved : Wednesday, October 01, 2025 at 02:55:42 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 4 y^{\prime \prime } y-5 {y^{\prime }}^{2}+a y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 34

ode:=4*diff(diff(y(x),x),x)*y(x)-5*diff(y(x),x)^2+a*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {16 \,{\mathrm e}^{\sqrt {a}\, x} a^{2}}{\left ({\mathrm e}^{\frac {\sqrt {a}\, x}{2}} c_1 -c_2 \right )^{4}} \\ \end{align*}

✓ Mathematica. Time used: 0.362 (sec). Leaf size: 46

ode=a*y[x]^2 - 5*D[y[x],x]^2 + 4*y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{a-K[1]^2}dK[1]\&\right ]\left [c_1-\frac {K[2]}{4}\right ]dK[2]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)**2 + 4*y(x)*Derivative(y(x), (x, 2)) - 5*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(5)*sqrt((a*y(x) + 4*Derivative(y(x), (x, 2)))*y(x))/5 + Derivative(y(x), x) cannot be solved by the factorable group method