15.18.41 problem 41

Internal problem ID [3284]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 41
Date solved : Sunday, March 30, 2025 at 01:31:21 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} \left (1-{\mathrm e}^{x}\right ) y^{\prime \prime }&={\mathrm e}^{x} y^{\prime } \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=1 \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 27

ode:=(1-exp(x))*diff(diff(y(x),x),x) = exp(x)*diff(y(x),x); 
ic:=y(1) = 0, D(y)(1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\left (\ln \left ({\mathrm e}^{x}\right )+\ln \left (-1+{\mathrm e}\right )-\ln \left (-1+{\mathrm e}^{x}\right )-1\right ) \left (-1+{\mathrm e}\right ) \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 27

ode=(1-Exp[x])*D[y[x],{x,2}]==Exp[x]*D[y[x],x]; 
ic={y[1]==0,Derivative[1][y][1]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -2 (e-1) \left (\text {arctanh}(1-2 e)-\text {arctanh}\left (1-2 e^x\right )\right ) \]

✓ Sympy. Time used: 0.274 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - exp(x))*Derivative(y(x), (x, 2)) - exp(x)*Derivative(y(x), x),0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (1 - e\right ) - \left (1 - e\right ) \log {\left (e^{x} - 1 \right )} - e \log {\left (-1 + e \right )} - 1 + \log {\left (-1 + e \right )} + e \]