54.7.15 problem 1605 (6.15)

Internal problem ID [12864]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1605 (6.15)
Date solved : Wednesday, October 01, 2025 at 02:22:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y}&=0 \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x)+a*exp(x)*y(x)^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : solve: Cannot solve a*sqrt(y(x))*exp(x) + Derivative(y(x), (x, 2))