87.22.20 problem 20

Internal problem ID [23765]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:44:59 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-6 y&=\cos \left (t \right )+57 \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=7 \\ \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-diff(y(t),t)-6*y(t) = cos(t)+57*sin(t); 
ic:=[y(0) = 1, D(y)(0) = 7]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \cos \left (t \right )-8 \sin \left (t \right )-3 \,{\mathrm e}^{-2 t}+3 \,{\mathrm e}^{3 t} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 26

ode=D[y[t],{t,2}]-D[y[t],{t,1}]-6*y[t]==Cos[t]+57*Sin[t]; 
ic={y[0]==1,Derivative[1][y][0] ==7}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 3 e^{-2 t} \left (e^{5 t}-1\right )-8 \sin (t)+\cos (t) \end{align*}

✓ Sympy. Time used: 0.157 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*y(t) - 57*sin(t) - cos(t) - Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 7} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 3 e^{3 t} - 8 \sin {\left (t \right )} + \cos {\left (t \right )} - 3 e^{- 2 t} \]