68.13.24 problem 41

Internal problem ID [17677]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number : 41
Date solved : Thursday, October 02, 2025 at 02:26:59 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 36

ode:=diff(diff(diff(y(t),t),t),t)-y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = {\mathrm e}^{t}-\sqrt {3}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )-{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 52

ode=D[ y[t],{t,3}]-y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t/2} \left (e^{3 t/2}-\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )-\cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 37

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

\[ y{\left (t \right )} = \left (- \sqrt {3} \sin {\left (\frac {\sqrt {3} t}{2} \right )} - \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} + e^{t} \]