87.22.35 problem 35

Internal problem ID [23780]
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 : 35
Date solved : Thursday, October 02, 2025 at 09:45:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+6 y&=250 \,{\mathrm e}^{t} \cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=-7 \\ \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+6*y(t) = 250*exp(t)*cos(t); 
ic:=[y(0) = 2, D(y)(0) = -7]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{t} \left (9 \sin \left (t \right )+13 \cos \left (t \right )\right )-19 \,{\mathrm e}^{-t}+8 \,{\mathrm e}^{-6 t} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 35

ode=D[y[t],{t,2}]+7*D[y[t],t]+6*y[t]==250*Exp[t]*Cos[t]; 
ic={y[0]==2,Derivative[1][y][0] ==-7}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-6 t} \left (8-19 e^{5 t}\right )+9 e^t \sin (t)+13 e^t \cos (t) \end{align*}

✓ Sympy. Time used: 0.181 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) - 250*exp(t)*cos(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -7} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (9 \sin {\left (t \right )} + 13 \cos {\left (t \right )}\right ) e^{t} - 19 e^{- t} + 8 e^{- 6 t} \]