70.22.3 problem 16

Internal problem ID [19070]
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 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 16
Date solved : Thursday, October 02, 2025 at 03:37:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+17 y&=g \left (t \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: 0.280 (sec). Leaf size: 31

ode:=4*diff(diff(y(t),t),t)+4*diff(y(t),t)+17*y(t) = g(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\int _{0}^{t}g \left (\textit {\_U1} \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\textit {\_U1}}{2}} \sin \left (-2 t +2 \textit {\_U1} \right )d \textit {\_U1}}{8} \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 141

ode=4*D[y[t],{t,2}]+4*D[y[t],t]+17*y[t]==g[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t/2} \left (-\sin (2 t) \int _1^0\frac {1}{8} e^{\frac {K[1]}{2}} \cos (2 K[1]) g(K[1])dK[1]+\sin (2 t) \int _1^t\frac {1}{8} e^{\frac {K[1]}{2}} \cos (2 K[1]) g(K[1])dK[1]+\cos (2 t) \left (\int _1^t-\frac {1}{8} e^{\frac {K[2]}{2}} g(K[2]) \sin (2 K[2])dK[2]-\int _1^0-\frac {1}{8} e^{\frac {K[2]}{2}} g(K[2]) \sin (2 K[2])dK[2]\right )\right ) \end{align*}

✓ Sympy. Time used: 6.816 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-g(t) + 17*y(t) + 4*Derivative(y(t), t) + 4*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)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\int g{\left (t \right )} e^{\frac {t}{2}} \sin {\left (2 t \right )}\, dt}{8} + \frac {\int \limits ^{0} g{\left (t \right )} e^{\frac {t}{2}} \sin {\left (2 t \right )}\, dt}{8}\right ) \cos {\left (2 t \right )} + \left (\frac {\int g{\left (t \right )} e^{\frac {t}{2}} \cos {\left (2 t \right )}\, dt}{8} - \frac {\int \limits ^{0} g{\left (t \right )} e^{\frac {t}{2}} \cos {\left (2 t \right )}\, dt}{8}\right ) \sin {\left (2 t \right )}\right ) e^{- \frac {t}{2}} \]