76.17.7 problem 16

Internal problem ID [17693]
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 : 16
Date solved : Tuesday, January 28, 2025 at 10:59:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=\frac {{\mathrm e}^{2 t}}{t^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 36

dsolve(diff(y(t),t$2)+4*diff(y(t),t)+4*y(t)=t^(-2)*exp(2*t),y(t), singsol=all)
 

\[ y = \left (-4 t +1\right ) {\mathrm e}^{-2 t} \operatorname {Ei}_{1}\left (-4 t \right )+\left (c_{1} t +c_{2} \right ) {\mathrm e}^{-2 t}-{\mathrm e}^{2 t} \]

✓ Solution by Mathematica

Time used: 0.257 (sec). Leaf size: 92

DSolve[D[y[t],{t,2}]+4*D[y[t],t]+2*y[t]==t^(-2)*Exp[2*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-\left (\left (2+\sqrt {2}\right ) t\right )} \left (\left (2 \sqrt {2}-1\right ) e^{2 \sqrt {2} t} \operatorname {ExpIntegralEi}\left (-\left (\left (-4+\sqrt {2}\right ) t\right )\right )-\left (1+2 \sqrt {2}\right ) \operatorname {ExpIntegralEi}\left (\left (4+\sqrt {2}\right ) t\right )+2 \left (c_2 e^{2 \sqrt {2} t}+c_1\right )\right ) \]