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 : Monday, January 27, 2025 at 07:30:55 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*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 27

dsolve([(1-exp(x))*diff(y(x),x$2)=exp(x)*diff(y(x),x),y(1) = 0, D(y)(1) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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