14.16.6 problem 20

Internal problem ID [2676]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 20
Date solved : Sunday, March 30, 2025 at 12:13:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.151 (sec). Leaf size: 43

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+y(t) = t^3; 
ic:=y(0) = 2, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {4 \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}+t^{3}-4 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )-3 t^{2}+6 \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 60

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

\[ y(t)\to t^3-3 t^2-\frac {4 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}-4 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+6 \]

✓ Sympy. Time used: 0.201 (sec). Leaf size: 49

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

\[ y{\left (t \right )} = t^{3} - 3 t^{2} + \left (- \frac {4 \sqrt {3} \sin {\left (\frac {\sqrt {3} t}{2} \right )}}{3} - 4 \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + 6 \]