14.18.8 problem 8

Internal problem ID [2692]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.11, Differential equations with discontinuous right-hand sides. Excercises page 243
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 02:37:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t^{2} & 0\le t <1 \\ 0 & 1\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 2.531 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)+y(t) = piecewise(0 <= t and t < 1,t^2,1 <= t,0); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = 2 \cos \left (t \right )+\left (\left \{\begin {array}{cc} t^{2}-2 & t <1 \\ 2 \sin \left (-1+t \right )-\cos \left (-1+t \right ) & 1\le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 49

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} t^2+2 \cos (t)-2 & 0<t\leq 1 \\ -\cos (1-t)+2 \cos (t)-2 \sin (1-t) & t>1 \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.354 (sec). Leaf size: 15

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

\[ y{\left (t \right )} = \begin {cases} t^{2} - 2 & \text {for}\: t \geq 0 \wedge t < 1 \\0 & \text {for}\: t \geq 1 \\\text {NaN} & \text {otherwise} \end {cases} + 2 \cos {\left (t \right )} \]