Internal problem ID [1110]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-3 y^{\prime }+2 y-\frac {1}{{\mathrm e}^{-x}+1}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve([diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=1/(1+exp(-x)),exp(2*x)],y(x), singsol=all)
\[ y \relax (x ) = \left ({\mathrm e}^{x} c_{1}-\ln \left ({\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{x}+\ln \left (1+{\mathrm e}^{x}\right ) \left (1+{\mathrm e}^{x}\right )-1+c_{2}\right ) {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 34
DSolve[y''[x]-3*y'[x]+2*y[x]==1/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^x \left (2 \left (e^x+1\right ) \tanh ^{-1}\left (2 e^x+1\right )+c_2 e^x-1+c_1\right ) \\ \end{align*}