87.22.22 problem 22

Internal problem ID [23767]
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 : 22
Date solved : Thursday, October 02, 2025 at 09:45:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+13 y^{\prime }+36 y&=10-72 t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+13*diff(y(t),t)+36*y(t) = 10-72*t; 
ic:=[y(0) = 2, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -2 t +2 \,{\mathrm e}^{-4 t}+1-{\mathrm e}^{-9 t} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+13*D[y[t],{t,1}]+36*y[t]==10-72*t; 
ic={y[0]==2,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -2 t-e^{-9 t}+2 e^{-4 t}+1 \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(72*t + 36*y(t) + 13*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 10,0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 2 t + 1 + 2 e^{- 4 t} - e^{- 9 t} \]