76.17.10 problem 19

Internal problem ID [17696]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 11:01:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 24

dsolve(diff(y(t),t$2)-2*diff(y(t),t)+y(t)=exp(t)/(1+t^2),y(t), singsol=all)
 

\[ y = {\mathrm e}^{t} \left (c_{2} +c_{1} t -\frac {\ln \left (t^{2}+1\right )}{2}+\arctan \left (t \right ) t \right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 35

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

\[ y(t)\to \frac {1}{2} e^t \left (2 t \arctan (t)-\log \left (t^2+1\right )+2 (c_2 t+c_1)\right ) \]