87.22.25 problem 25

Internal problem ID [23770]
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 : 25
Date solved : Thursday, October 02, 2025 at 09:45:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+10 y&=3 \,{\mathrm e}^{-2 t}-6 \,{\mathrm e}^{-5 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+10*y(t) = 3*exp(-2*t)-6*exp(-5*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+7*D[y[t],{t,1}]+10*y[t]==3*Exp[-2*t]-6*Exp[-5*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-5 t} \left (e^{3 t} (t-1)+2 t+1\right ) \end{align*}

✓ Sympy. Time used: 0.256 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(10*y(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 3*exp(-2*t) + 6*exp(-5*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t + \left (2 t + 1\right ) e^{- 3 t} - 1\right ) e^{- 2 t} \]