87.22.31 problem 31

Internal problem ID [23776]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 31
Date solved : Thursday, October 02, 2025 at 09:45:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=3 \,{\mathrm e}^{2 t} t -4 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+4*y(t) = 3*exp(2*t)*t-4; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 24

ode=D[y[t],{t,2}]-4*D[y[t],t]+4*y[t]==3*t*Exp[2*t]-4; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} e^{2 t} \left (t^3-4 t+2\right )-1 \end{align*}

✓ Sympy. Time used: 0.210 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*t*exp(2*t) + 4*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 4,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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