67.3.12 problem Problem 13

Internal problem ID [14030]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 13
Date solved : Tuesday, January 28, 2025 at 06:13:01 AM
CAS classification : [[_high_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.677 (sec). Leaf size: 47

dsolve([diff(y(t),t$4)+y(t)=0,y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 1/2*2^(1/2)],y(t), singsol=all)
 

\[ y = -\frac {\sinh \left (\frac {\sqrt {2}\, t}{2}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{2}+\frac {\cosh \left (\frac {\sqrt {2}\, t}{2}\right ) \left (2 \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 61

DSolve[{D[y[t],{t,4}]+y[t]==0,{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==1/Sqrt[2]}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{4} e^{-\frac {t}{\sqrt {2}}} \left (\left (e^{\sqrt {2} t}+1\right ) \sin \left (\frac {t}{\sqrt {2}}\right )-\left (e^{\sqrt {2} t}-1\right ) \cos \left (\frac {t}{\sqrt {2}}\right )\right ) \]