14.21.10 problem 10

Internal problem ID [2719]
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 : 10
Date solved : Monday, January 27, 2025 at 06:12:16 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 71

dsolve(diff(y(t),t$4)-y(t)=g(t),y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 118

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

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