67.18.4 problem 27.1 (d)

Internal problem ID [16875]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (d)
Date solved : Thursday, October 02, 2025 at 01:40:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=t^{3} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)-4*y(t) = t^3; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {3 t}{8}-\frac {t^{3}}{4}+\frac {13 \,{\mathrm e}^{-2 t}}{32}+\frac {19 \,{\mathrm e}^{2 t}}{32} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 34

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

\begin{align*} y(t)&\to \frac {1}{32} \left (-4 t \left (2 t^2+3\right )-11 e^{-2 t}+43 e^{2 t}\right ) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**3 - 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {t^{3}}{4} - \frac {3 t}{8} + \frac {19 e^{2 t}}{32} + \frac {13 e^{- 2 t}}{32} \]