87.22.30 problem 30

Internal problem ID [23775]
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 : 30
Date solved : Thursday, October 02, 2025 at 09:45:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=8 \,{\mathrm e}^{t}+5 t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 8*exp(t)+5*t; 
ic:=[y(0) = 3, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {2}{5}+t +{\mathrm e}^{t}+\frac {6 \,{\mathrm e}^{-t} \left (2 \cos \left (2 t \right )+\sin \left (2 t \right )\right )}{5} \]

✓ Mathematica. Time used: 0.338 (sec). Leaf size: 39

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==8*Exp[t]+5*t; 
ic={y[0]==3,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t+e^t+\frac {6}{5} e^{-t} \sin (2 t)+\frac {12}{5} e^{-t} \cos (2 t)-\frac {2}{5} \end{align*}

✓ Sympy. Time used: 0.165 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-5*t + 5*y(t) - 8*exp(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t + \left (\frac {6 \sin {\left (2 t \right )}}{5} + \frac {12 \cos {\left (2 t \right )}}{5}\right ) e^{- t} + e^{t} - \frac {2}{5} \]