Internal problem ID [4677]
Book: Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill
2014
Section: Chapter 12. VARIATION OF PARAMETERS. page 104
Problem number: Problem 12.2.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_y]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime }-\frac {{\mathrm e}^{x}}{1+{\mathrm e}^{-x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 65
dsolve(diff(y(x),x$3)-3*diff(y(x),x$2)+2*diff(y(x),x)=exp(x)/(1+exp(-x)),y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{x}+{\mathrm e}^{x} \ln \left ({\mathrm e}^{-x}\right )+\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{2 x} c_{1}}{2}-\frac {x}{2}-\frac {\ln \left (1+{\mathrm e}^{-x}\right )}{2}-\frac {{\mathrm e}^{2 x} \ln \left (1+{\mathrm e}^{-x}\right )}{2}-\ln \left (1+{\mathrm e}^{-x}\right ) {\mathrm e}^{x}+c_{3} \]
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 59
DSolve[y'''[x]-3*y''[x]+2*y'[x]==Exp[x]/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \left (2 e^x+1\right ) \log \left (e^x+1\right )-e^{2 x} \tanh ^{-1}\left (2 e^x+1\right )+\frac {1}{2} e^x \left (c_2 e^x+1+2 c_1\right )+c_3 \\ \end{align*}