67.4.8 problem Problem 2(h)

Internal problem ID [13981]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(h)
Date solved : Monday, March 31, 2025 at 08:20:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=2+t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+5*y(t) = t+2; 
ic:=y(0) = 4, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {12}{25}+\frac {t}{5}+\frac {2 \,{\mathrm e}^{t} \left (44 \cos \left (2 t \right )-17 \sin \left (2 t \right )\right )}{25} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 32

ode=D[y[t],{t,2}]-2*D[y[t],t]+5*y[t]==2+t; 
ic={y[0]==4,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{25} \left (5 t-34 e^t \sin (2 t)+88 e^t \cos (2 t)+12\right ) \]

✓ Sympy. Time used: 0.218 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 5*y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 2,0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t}{5} + \left (- \frac {34 \sin {\left (2 t \right )}}{25} + \frac {88 \cos {\left (2 t \right )}}{25}\right ) e^{t} + \frac {12}{25} \]