87.22.23 problem 23

Internal problem ID [23768]
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 : 23
Date solved : Thursday, October 02, 2025 at 09:45:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-15 y&=16 t \,{\mathrm e}^{-t}-15 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-9 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)-15*y(t) = 16*t*exp(-t)-15; 
ic:=[y(0) = 1, D(y)(0) = -9]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 1-t \,{\mathrm e}^{-t}+{\mathrm e}^{-5 t}-{\mathrm e}^{3 t} \]

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 27

ode=D[y[t],{t,2}]+2*D[y[t],{t,1}]-15*y[t]==16*t*Exp[-t]-15; 
ic={y[0]==1,Derivative[1][y][0] ==-9}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.195 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-16*t*exp(-t) - 15*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 15,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -9} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - t e^{- t} - e^{3 t} + 1 + e^{- 5 t} \]