76.15.36 problem 38

Internal problem ID [17598]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 38
Date solved : Thursday, March 13, 2025 at 10:37:45 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\left \{\begin {array}{cc} 1 & 0\le t \le \frac {\pi }{2} \\ 0 & \frac {\pi }{2}<t \end {array}\right . \end{align*}

With initial conditions

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

✓ Maple. Time used: 21.647 (sec). Leaf size: 64

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = piecewise(0 <= t and t <= 1/2*Pi,1,1/2*Pi < t,0); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {\left (\left \{\begin {array}{cc} 0 & t <0 \\ -2+{\mathrm e}^{-t} \left (2 \cos \left (2 t \right )+\sin \left (2 t \right )\right ) & t <\frac {\pi }{2} \\ \left ({\mathrm e}^{-t}+{\mathrm e}^{\frac {\pi }{2}-t}\right ) \left (2 \cos \left (2 t \right )+\sin \left (2 t \right )\right ) & \frac {\pi }{2}\le t \end {array}\right .\right )}{10} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 78

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {1}{10} e^{-t} \left (-2 \cos (2 t)+2 e^t-\sin (2 t)\right ) & t>0\land 2 t\leq \pi \\ -\frac {1}{10} e^{-t} \left (1+e^{\pi /2}\right ) (2 \cos (2 t)+\sin (2 t)) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.489 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t <= pi/2)), (0, t > pi/2)) + 5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = \begin {cases} \frac {C_{1}}{e^{\frac {5 t}{2}} + 1} + \frac {e^{\frac {5 t}{2}}}{5 e^{\frac {5 t}{2}} + 5} & \text {for}\: t \leq \frac {\pi }{2} \wedge t > 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (t \right )} = \begin {cases} \frac {C_{1}}{e^{\frac {5 t}{2} - \frac {5 \pi }{4}} + e^{\frac {5 \pi }{4}}} & \text {for}\: t > \frac {\pi }{2} \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]