87.22.9 problem 9

Internal problem ID [23754]
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 : 9
Date solved : Thursday, October 02, 2025 at 09:44:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+10 y^{\prime }+25 y&=2 \,{\mathrm e}^{-5 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)+10*diff(y(t),t)+25*y(t) = 2*exp(-5*t); 
ic:=[y(0) = 0, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-5 t} t \left (t -1\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 15

ode=D[y[t],{t,2}]+10*D[y[t],t]+25*y[t]==2*Exp[-5*t]; 
ic={y[0]==0,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-5 t} (t-1) t \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(25*y(t) + 10*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 2*exp(-5*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t \left (t - 1\right ) e^{- 5 t} \]