77.1.79 problem 98 (page 135)

Internal problem ID [17890]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 98 (page 135)
Date solved : Thursday, March 13, 2025 at 11:08:58 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 2.173 (sec). Leaf size: 34

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 {c_{1}^{2} {\mathrm e}^{x}+1}{c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 7.257 (sec). Leaf size: 163

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

\begin{align*} y(x)\to -\frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {x-c_1}{2}\right )}} \\ y(x)\to \frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {x-c_1}{2}\right )}} \\ y(x)\to -\frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {-x+c_1}{2}\right )}} \\ y(x)\to \frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {-x+c_1}{2}\right )}} \\ 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