60.1.378 problem 387

Internal problem ID [10392]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 387
Date solved : Sunday, March 30, 2025 at 04:34:35 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.398 (sec). Leaf size: 119

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

\begin{align*} \frac {2 \ln \left (y\right ) y+4 \,\operatorname {arctanh}\left (\sqrt {{\mathrm e}^{x} \left ({\mathrm e}^{x}+4 y\right )}\, {\mathrm e}^{-x}\right ) y-c_1 y-{\mathrm e}^{x}+\sqrt {{\mathrm e}^{x} \left ({\mathrm e}^{x}+4 y\right )}}{y} &= 0 \\ \frac {2 \ln \left (y\right ) y-4 \,\operatorname {arctanh}\left (\sqrt {{\mathrm e}^{x} \left ({\mathrm e}^{x}+4 y\right )}\, {\mathrm e}^{-x}\right ) y-c_1 y-{\mathrm e}^{x}-\sqrt {{\mathrm e}^{x} \left ({\mathrm e}^{x}+4 y\right )}}{y} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 4.672 (sec). Leaf size: 195

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

\begin{align*} \text {Solve}\left [\log (y(x))-\frac {-\frac {4 \sqrt {\frac {e^x}{y(x)}+4} y(x)^{3/2} \text {arcsinh}\left (\frac {e^{x/2}}{2 \sqrt {y(x)}}\right )}{\sqrt {4 y(x)+e^x}}-e^{x/2} \sqrt {4 y(x)+e^x}+e^x}{2 y(x)}&=c_1,y(x)\right ] \\ \text {Solve}\left [\log (y(x))-\frac {\frac {4 \sqrt {\frac {e^x}{y(x)}+4} y(x)^{3/2} \text {arcsinh}\left (\frac {e^{x/2}}{2 \sqrt {y(x)}}\right )}{\sqrt {4 y(x)+e^x}}+e^{x/2} \sqrt {4 y(x)+e^x}+e^x}{2 y(x)}&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \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((4*y(x) + exp(x))*exp(x))/2 + exp(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method