14.11.2 problem 2

Internal problem ID [2595]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.5. Method of judicious guessing. Excercises page 164
Problem number : 2
Date solved : Monday, January 27, 2025 at 06:01:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 39

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

\[ y = \frac {\left (\alpha +2\right )^{3} \left (c_1 t +c_2 \right ) {\mathrm e}^{-2 t}+\left (\alpha t +2 t -2\right ) {\mathrm e}^{\alpha t}}{\left (\alpha +2\right )^{3}} \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 37

DSolve[D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==t*Exp[\[Alpha]*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {e^{\alpha t} ((\alpha +2) t-2)}{(\alpha +2)^3}+e^{-2 t} (c_2 t+c_1) \]