74.13.24 problem 41

Internal problem ID [16390]
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 : Tuesday, January 28, 2025 at 09:07:25 AM
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*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 36

dsolve([diff(y(t),t$3)-y(t)=0,y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 3],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 52

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

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