70.20.14 problem 16 (c.1)

Internal problem ID [19048]
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.1)
Date solved : Thursday, October 02, 2025 at 03:37:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{4}+u&=\frac {\left (\left \{\begin {array}{cc} 1 & \frac {3}{2}\le t <\frac {5}{2} \\ 0 & \operatorname {otherwise} \end {array}\right .\right )}{2} \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: 0.704 (sec). Leaf size: 181

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

\[ u = \left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ \frac {\left (i \sqrt {7}+21\right ) \left (3 i {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}} \sqrt {7}-3 i \sqrt {7}-32 \,{\mathrm e}^{-\frac {3 \left (i \sqrt {7}+\frac {1}{3}\right ) \left (t -\frac {3}{2}\right )}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+63\right )}{2688} & t <\frac {5}{2} \\ \frac {\left (3 i {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (2 t +1\right )}{8}} \sqrt {7}-3 i \sqrt {7}\, {\mathrm e}^{\frac {1}{8}+\frac {3 i \sqrt {7}\, t}{4}}-31 \,{\mathrm e}^{\frac {3 i \sqrt {7}\, \left (2 t +1\right )}{8}}+32 \,{\mathrm e}^{\frac {1}{8}+\frac {15 i \sqrt {7}}{8}}-32 \,{\mathrm e}^{\frac {3 i \sqrt {7}}{2}}+31 \,{\mathrm e}^{\frac {1}{8}+\frac {3 i \sqrt {7}\, t}{4}}\right ) \left (i \sqrt {7}+21\right ) {\mathrm e}^{\frac {3}{16}-\frac {3 i \left (5+2 t \right ) \sqrt {7}}{16}-\frac {t}{8}}}{2688} & \frac {5}{2}\le t \end {array}\right . \]

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 198

ode=D[u[t],{t,2}]+1/4*D[u[t],t]+u[t]==1/2*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]
 

\begin{align*} u(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{42} \left (-21 e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+\sqrt {7} e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21\right ) & \frac {3}{2}<t\leq \frac {5}{2} \\ \frac {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 )}{6 \sqrt {7}} & 2 t>5 \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 2.213 (sec). Leaf size: 7

from sympy import * 
t = symbols("t") 
u = Function("u") 
ode = Eq(-Piecewise((1, (t >= 3/2) & (t < 5/2)), (0, True))/2 + 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)
 

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