70.20.9 problem 9

Internal problem ID [19043]
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 : 9
Date solved : Thursday, October 02, 2025 at 03:37:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} \frac {t}{2} & 0\le t <6 \\ 3 & 6\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=6 \\ y^{\prime }\left (0\right )&=8 \\ \end{align*}

✓ Maple. Time used: 0.385 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+y(t) = piecewise(0 <= t and t < 6,1/2*t,6 <= t,3); 
ic:=[y(0) = 6, D(y)(0) = 8]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 6 \cos \left (t \right )+\frac {15 \sin \left (t \right )}{2}+\frac {\left (\left \{\begin {array}{cc} t & t <6 \\ 6+\sin \left (t -6\right ) & 6\le t \end {array}\right .\right )}{2} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 63

ode=D[y[t],{t,2}]+y[t]==Piecewise[{  {t/2,0<= t <6}, {3,t>=6}}]; 
ic={y[0]==6,Derivative[1][y][0] ==8}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 6 \cos (t)+8 \sin (t) & t\leq 0 \\ \frac {1}{2} (t+12 \cos (t)+15 \sin (t)) & 0<t\leq 6 \\ 6 \cos (t)-\frac {1}{2} \sin (6-t)+\frac {15 \sin (t)}{2}+3 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.212 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((t/2, (t >= 0) & (t < 6)), (3, t >= 6)) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 6, Subs(Derivative(y(t), t), t, 0): 8} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {t}{2} & \text {for}\: t \geq 0 \wedge t < 6 \\3 & \text {for}\: t \geq 6 \\\text {NaN} & \text {otherwise} \end {cases} + \frac {15 \sin {\left (t \right )}}{2} + 6 \cos {\left (t \right )} \]