68.9.29 problem 47

Internal problem ID [17492]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 47
Date solved : Thursday, October 02, 2025 at 02:24:42 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+b y^{\prime }+c y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\cos \left (t \right )}{t^{3}} \end{align*}

With initial conditions

\begin{align*} y \left (\pi \right )&=0 \\ y \left (2 \pi \right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 5

ode:=diff(diff(y(t),t),t)+b*diff(y(t),t)+c*y(t) = 0; 
ic:=[y(Pi) = 0, y(2*Pi) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 0 \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 109

ode=D[y[t],{t,2}]+b*D[y[t],t]+c*y[t]==0; 
ic={y[Pi]==0,y[2*Pi]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{\sqrt {b^2-4 c} \pi -\frac {1}{2} \left (b+\sqrt {b^2-4 c}\right ) t} \left (-1+e^{\sqrt {b^2-4 c} (t-\pi )}\right ) c_1 & e^{-\frac {1}{2} \left (3 b+\sqrt {b^2-4 c}\right ) \pi } \left (-1+e^{\sqrt {b^2-4 c} \pi }\right )=0 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.172 (sec). Leaf size: 3

from sympy import * 
t = symbols("t") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(b*Derivative(y(t), t) + c*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(pi): 0, y(2*pi): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 0 \]