23.2.60 problem 62

Internal problem ID [5415]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 62
Date solved : Tuesday, September 30, 2025 at 12:40:01 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x}&=0 \end{align*}

✓ Maple. Time used: 0.355 (sec). Leaf size: 30

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

\begin{align*} y &= -2 \,{\mathrm e}^{\frac {x}{2}} \\ y &= 2 \,{\mathrm e}^{\frac {x}{2}} \\ y &= \frac {1}{c_{1}}+c_{1} {\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 6.781 (sec). Leaf size: 85

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

\begin{align*} y(x)&\to -\sqrt {-e^{-c_1} \left (-e^x+e^{c_1}\right ){}^2}\\ y(x)&\to \sqrt {-e^{-c_1} \left (e^x-e^{c_1}\right ){}^2}\\ y(x)&\to -2 e^{x/2}\\ y(x)&\to 2 e^{x/2} \end{align*}

✗ Sympy

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

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