23.4.76 problem 76

Internal problem ID [6378]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 76
Date solved : Friday, October 03, 2025 at 02:05:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 35

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

\[ y = -\ln \left (2\right )+\ln \left (-\frac {\left (c_1 -4\right ) \sec \left (-\frac {\ln \left (x \right ) \sqrt {c_1 -4}}{2}+c_2 \right )^{2}}{a \,x^{2}}\right ) \]

✓ Mathematica. Time used: 60.979 (sec). Leaf size: 85

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

\begin{align*} y(x)&\to \log \left (\frac {(2+a c_1) \text {sech}^2\left (\sqrt {1+\frac {a c_1}{2}} (-\log (x)+c_2)\right )}{a x^2}\right )\\ y(x)&\to \log \left (\frac {(2+a c_1) \text {sech}^2\left (\sqrt {1+\frac {a c_1}{2}} (-\log (x)+c_2)\right )}{a x^2}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x*(-a*exp(y(x)) - Derivative(y(x), (x, 2))) + Derivative(y(x),