76.20.16 problem 16 (c.3)

Internal problem ID [17685]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 16 (c.3)
Date solved : Thursday, March 13, 2025 at 10:47:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{4}+u&=2 \left (\left \{\begin {array}{cc} 1 & \frac {3}{2}\le t <\frac {5}{2} \\ 0 & \operatorname {otherwise} \end {array}\right .\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 17.671 (sec). Leaf size: 184

ode:=diff(diff(u(t),t),t)+1/4*diff(u(t),t)+u(t) = 2*piecewise(3/2 <= t and t < 5/2,1,0); 
ic:=u(0) = 0, D(u)(0) = 0; 
dsolve([ode,ic],u(t),method='laplace');
 

\[ u = \frac {\left (i \sqrt {7}+21\right ) \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}} \sqrt {7}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (3-2 t \right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (-5+2 t \right )}{16}+\frac {5}{16}-\frac {t}{8}}+3 i {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}} \sqrt {7}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+32 \,{\mathrm e}^{-\frac {3 i \sqrt {7}\, \left (-5+2 t \right )}{16}+\frac {5}{16}-\frac {t}{8}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (3-2 t \right ) \sqrt {7}}{16}-\frac {t}{8}} & \frac {5}{2}\le t \end {array}\right .\right )}{672} \]

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 197

ode=D[u[t],{t,2}]+1/4*D[u[t],t]+u[t]==2*Piecewise[{  {1,3/2<= t <5/2}, {0,True}}]; 
ic={u[0]==0,Derivative[1][u][0] ==0}; 
DSolve[{ode,ic},u[t],t,IncludeSingularSolutions->True]
 

\[ u(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -2 e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+\frac {2 e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )}{3 \sqrt {7}}+2 & \frac {3}{2}<t\leq \frac {5}{2} \\ \frac {2 e^{\frac {3}{16}-\frac {t}{8}} \left (-3 \sqrt {7} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+3 \sqrt {7} \sqrt [8]{e} \cos \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )+\sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )-\sqrt [8]{e} \sin \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )\right )}{3 \sqrt {7}} & 2 t>5 \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 2.838 (sec). Leaf size: 7

from sympy import * 
t = symbols("t") 
u = Function("u") 
ode = Eq(-2*Piecewise((1, (t >= 3/2) & (t < 5/2)), (0, True)) + u(t) + Derivative(u(t), t)/4 + Derivative(u(t), (t, 2)),0) 
ics = {u(0): 0, Subs(Derivative(u(t), t), t, 0): 0} 
dsolve(ode,func=u(t),ics=ics)
 

\[ u{\left (t \right )} = \begin {cases} 0 & \text {for}\: t < \frac {3}{2} \\\text {NaN} & \text {otherwise} \end {cases} \]