12.9.4 problem 4

Internal problem ID [1760]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 4
Date solved : Monday, January 27, 2025 at 05:34:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=\frac {1}{1+{\mathrm e}^{-x}} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 34

dsolve([diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=1/(1+exp(-x)),exp(2*x)],singsol=all)
 

\[ y = {\mathrm e}^{x} \left (\ln \left (1+{\mathrm e}^{x}\right ) \left (1+{\mathrm e}^{x}\right )+\left (-1-{\mathrm e}^{x}\right ) \ln \left ({\mathrm e}^{x}\right )+{\mathrm e}^{x} c_1 +c_2 -1\right ) \]

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 34

DSolve[D[y[x],{x,2}]-3*D[y[x],x]+2*y[x]==1/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x \left (2 \left (e^x+1\right ) \text {arctanh}\left (2 e^x+1\right )+c_2 e^x-1+c_1\right ) \]