50.25.8 problem 4(d)

Internal problem ID [8180]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section A, Drill exercises. Page 309
Problem number : 4(d)
Date solved : Monday, January 27, 2025 at 03:45:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-7 y^{\prime }+12 y&=t \,{\mathrm e}^{2 t} \end{align*}

Using Laplace method

✓ Solution by Maple

Time used: 0.633 (sec). Leaf size: 47

dsolve(diff(y(t),t$2)-7*diff(y(t),t)+12*y(t)=t*exp(2*t),y(t), singsol=all)
 

\[ y = \frac {\left (2 t +3\right ) {\mathrm e}^{2 t}}{4}+\left (4 y \left (0\right )-y^{\prime }\left (0\right )-1\right ) {\mathrm e}^{3 t}+\frac {{\mathrm e}^{4 t} \left (-12 y \left (0\right )+4 y^{\prime }\left (0\right )+1\right )}{4} \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 35

DSolve[D[y[t],{t,2}]-7*D[y[t],t]+12*y[t]==t*Exp[2*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{4} e^{2 t} \left (2 t+4 c_1 e^t+4 c_2 e^{2 t}+3\right ) \]