Internal problem ID [5185]
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]]
\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime }=\frac {{\mathrm e}^{x}}{1+{\mathrm e}^{-x}}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 56
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)
\[ y \left (x \right ) = \frac {\left (-2 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-1\right ) \ln \left (1+{\mathrm e}^{-x}\right )}{2}+\frac {\left (2 \,{\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{-x}\right )}{2}+\frac {{\mathrm e}^{2 x} c_{1}}{2}+\frac {\left (2 c_{2} +1\right ) {\mathrm e}^{x}}{2}+c_{3} \]
✓ Solution by Mathematica
Time used: 0.137 (sec). Leaf size: 59
DSolve[y'''[x]-3*y''[x]+2*y'[x]==Exp[x]/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \left (-2 e^{2 x} \text {arctanh}\left (2 e^x+1\right )-\left (2 e^x+1\right ) \log \left (e^x+1\right )+e^x \left (c_2 e^x+1+2 c_1\right )\right )+c_3 \]