85.70.4 problem 5 (d)

Internal problem ID [22928]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 6. Solution of linear differential equations by Laplace transform. A Exercises at page 283
Problem number : 5 (d)
Date solved : Thursday, October 02, 2025 at 09:16:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+y(t) = 12*t; 
ic:=[y(0) = 4, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 24+12 t +\left (9 t -20\right ) {\mathrm e}^{t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 20

ode=D[y[t],{t,2}]-2*D[y[t],t]+y[t]==12*t; 
ic={y[0]==4,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 12 (t+2)+e^t (9 t-20) \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-12*t + y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 12 t + \left (9 t - 20\right ) e^{t} + 24 \]