46.6.10 problem 30

Internal problem ID [9631]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 06:21:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=1+t \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 26

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

\[ y = \frac {7}{25}+\frac {t}{5}+\frac {\left (-7 \cos \left (2 t \right )+51 \sin \left (2 t \right )\right ) {\mathrm e}^{t}}{25} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to \frac {1}{25} \left (5 t+51 e^t \sin (2 t)-7 e^t \cos (2 t)+7\right ) \end{align*}

✓ Sympy. Time used: 0.139 (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)) - 1,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t}{5} + \left (\frac {51 \sin {\left (2 t \right )}}{25} - \frac {7 \cos {\left (2 t \right )}}{25}\right ) e^{t} + \frac {7}{25} \]