85.67.19 problem 19

Internal problem ID [22910]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 216
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:16:31 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} i^{\prime \prime \prime \prime }+9 i^{\prime \prime }&=20 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(i(t),t),t),t),t)+9*diff(diff(i(t),t),t) = 20*exp(-t); 
ic:=[i(0) = 0, D(i)(0) = 0]; 
dsolve([ode,op(ic)],i(t), singsol=all);
 

\[ i = \left (-c_4 -2\right ) \cos \left (3 t \right )+\frac {\left (2-c_3 \right ) \sin \left (3 t \right )}{3}+c_3 t +c_4 +2 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 44

ode=D[i[t],{t,4}]+9*D[i[t],{t,2}]==20*Exp[-t]; 
ic={i[0]==0,Derivative[1][i][0] ==0}; 
DSolve[{ode,ic},i[t],t,IncludeSingularSolutions->True]
 

\begin{align*} i(t)&\to \frac {1}{9} \left (18 t+18 e^{-t}+3 c_2 t-c_1 \cos (3 t)-c_2 \sin (3 t)-18+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
i = Function("i") 
ode = Eq(9*Derivative(i(t), (t, 2)) + Derivative(i(t), (t, 4)) - 20*exp(-t),0) 
ics = {i(0): 0, Subs(Derivative(i(t), t), t, 0): 0} 
dsolve(ode,func=i(t),ics=ics)
 

\[ i{\left (t \right )} = C_{3} \sin {\left (3 t \right )} + C_{4} \cos {\left (3 t \right )} - C_{4} + t \left (2 - 3 C_{3}\right ) - 2 + 2 e^{- t} \]