87.22.26 problem 26

Internal problem ID [23771]
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 : 26
Date solved : Thursday, October 02, 2025 at 09:45:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }-3 y^{\prime }-y&=34 \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 23

ode:=4*diff(diff(y(t),t),t)-3*diff(y(t),t)-y(t) = 34*sin(t); 
ic:=[y(0) = 1, D(y)(0) = -2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 3 \cos \left (t \right )-5 \sin \left (t \right )+2 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{-\frac {t}{4}} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 75

ode=D[y[t],{t,2}]-3*D[y[t],{t,1}]-y[t]==34*Sin[t]; 
ic={y[0]==1,Derivative[1][y][0] ==-2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{26} \left (e^{-\frac {1}{2} \left (\sqrt {13}-3\right ) t} \left (\left (27 \sqrt {13}-89\right ) e^{\sqrt {13} t}-136 e^{\frac {1}{2} \left (\sqrt {13}-3\right ) t} \sin (t)-27 \sqrt {13}-89\right )+204 \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - 34*sin(t) - 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {27 \sqrt {13}}{26} - \frac {89}{26}\right ) e^{\frac {t \left (3 - \sqrt {13}\right )}{2}} + \left (- \frac {89}{26} + \frac {27 \sqrt {13}}{26}\right ) e^{\frac {t \left (3 + \sqrt {13}\right )}{2}} - \frac {68 \sin {\left (t \right )}}{13} + \frac {102 \cos {\left (t \right )}}{13} \]