70.19.8 problem 8

Internal problem ID [19018]
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 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 8
Date solved : Thursday, October 02, 2025 at 03:37:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=\cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 23

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

\[ y = \frac {\cos \left (t \right ) \left (1+4 \,{\mathrm e}^{t}\right )}{5}-\frac {2 \sin \left (t \right ) \left (1+{\mathrm e}^{t}\right )}{5} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 110

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

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

✓ Sympy. Time used: 0.135 (sec). Leaf size: 31

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

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