8.16.5 problem 19

Internal problem ID [2675]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 05:49:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+7 y&=\cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 42

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+7*y(t) = cos(t); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {52 \,{\mathrm e}^{-\frac {3 t}{2}} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{285}-\frac {2 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )}{15}+\frac {2 \cos \left (t \right )}{15}+\frac {\sin \left (t \right )}{15} \]

✓ Mathematica. Time used: 0.749 (sec). Leaf size: 63

ode=D[y[t],{t,2}]+3*D[y[t],t]+7*y[t]==Cos[t]; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{285} \left (19 \sin (t)+52 \sqrt {19} e^{-3 t/2} \sin \left (\frac {\sqrt {19} t}{2}\right )+38 \cos (t)-38 e^{-3 t/2} \cos \left (\frac {\sqrt {19} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.257 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(7*y(t) - cos(t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {52 \sqrt {19} \sin {\left (\frac {\sqrt {19} t}{2} \right )}}{285} - \frac {2 \cos {\left (\frac {\sqrt {19} t}{2} \right )}}{15}\right ) e^{- \frac {3 t}{2}} + \frac {\sin {\left (t \right )}}{15} + \frac {2 \cos {\left (t \right )}}{15} \]