14.21.11 problem 11

Internal problem ID [2720]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 02:40:18 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+y&=g \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 257

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+y(t) = g(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+\frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{4}-\frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+\frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+c_1 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )+c_2 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )+c_4 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \]

✓ Mathematica. Time used: 0.324 (sec). Leaf size: 331

ode=D[y[t],{t,4}]+y[t]==g[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-\frac {t}{\sqrt {2}}} \left (e^{\sqrt {2} t} \cos \left (\frac {t}{\sqrt {2}}\right ) \int _1^t-\frac {e^{-\frac {K[1]}{\sqrt {2}}} g(K[1]) \left (\cos \left (\frac {K[1]}{\sqrt {2}}\right )+\sin \left (\frac {K[1]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[1]+\cos \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{\frac {K[2]}{\sqrt {2}}} g(K[2]) \left (\cos \left (\frac {K[2]}{\sqrt {2}}\right )-\sin \left (\frac {K[2]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[2]+\sin \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{\frac {K[3]}{\sqrt {2}}} g(K[3]) \left (\cos \left (\frac {K[3]}{\sqrt {2}}\right )+\sin \left (\frac {K[3]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[3]+e^{\sqrt {2} t} \sin \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{-\frac {K[4]}{\sqrt {2}}} g(K[4]) \left (\cos \left (\frac {K[4]}{\sqrt {2}}\right )-\sin \left (\frac {K[4]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[4]+c_1 e^{\sqrt {2} t} \cos \left (\frac {t}{\sqrt {2}}\right )+c_2 \cos \left (\frac {t}{\sqrt {2}}\right )+c_3 \sin \left (\frac {t}{\sqrt {2}}\right )+c_4 e^{\sqrt {2} t} \sin \left (\frac {t}{\sqrt {2}}\right )\right ) \]

✓ Sympy. Time used: 6.745 (sec). Leaf size: 192

from sympy import * 
t = symbols("t") 
y = Function("y") 
g = Function("g") 
ode = Eq(-g(t) + y(t) + Derivative(y(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (C_{1} - \frac {\int g{\left (t \right )} e^{- \frac {\sqrt {2} t}{2}} \sin {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \cos {\left (\frac {\sqrt {2} t}{2} \right )} + \left (C_{2} + \frac {\int g{\left (t \right )} e^{- \frac {\sqrt {2} t}{2}} \cos {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \sin {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{\frac {\sqrt {2} t}{2}} + \left (\left (C_{3} + \frac {\int g{\left (t \right )} e^{\frac {\sqrt {2} t}{2}} \sin {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \sin {\left (\frac {\sqrt {2} t}{2} \right )} + \left (C_{4} + \frac {\int g{\left (t \right )} e^{\frac {\sqrt {2} t}{2}} \cos {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \cos {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{- \frac {\sqrt {2} t}{2}} \]