61.3.12 problem Problem 13

Internal problem ID [15311]
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 : Thursday, October 02, 2025 at 10:11:32 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*}

✓ Maple. Time used: 0.169 (sec). Leaf size: 47

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 1/2*2^(1/2)]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 61

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 71

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): sqrt(2)/2} 
dsolve(ode,func=y(t),ics=ics)
 

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