67.18.2 problem 27.1 (b)

Internal problem ID [16873]
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 (b)
Date solved : Thursday, October 02, 2025 at 01:40:01 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 25

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

\[ y = -\frac {t^{3}}{2}-\frac {3 t}{4}-\frac {3 t^{2}}{4}+\frac {35 \,{\mathrm e}^{2 t}}{8}-\frac {3}{8} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 31

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

\begin{align*} y(t)&\to e^{-4 t} \left (\int _0^te^{4 K[1]} K[1]^3dK[1]+4\right ) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = - \frac {t^{3}}{2} - \frac {3 t^{2}}{4} - \frac {3 t}{4} + \frac {35 e^{2 t}}{8} - \frac {3}{8} \]