44.25.8 problem 4(d)

Internal problem ID [9466]
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 : Tuesday, September 30, 2025 at 06:19:06 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

✓ Maple. Time used: 0.109 (sec). Leaf size: 47

ode:=diff(diff(y(t),t),t)-7*diff(y(t),t)+12*y(t) = t*exp(2*t); 
dsolve(ode,y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 35

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.153 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(2*t) + 12*y(t) - 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} e^{t} + C_{2} e^{2 t} + \frac {t}{2} + \frac {3}{4}\right ) e^{2 t} \]