75.8.8 problem 206

Internal problem ID [16753]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 206
Date solved : Monday, March 31, 2025 at 03:15:54 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: 0.363 (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: 11.076 (sec). Leaf size: 163

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 -\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