14.21.9 problem 9

Internal problem ID [2718]
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 : 9
Date solved : Monday, January 27, 2025 at 06:12:15 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 71

dsolve(diff(y(t),t$3)+diff(y(t),t)=tan(t),y(t), singsol=all)
 

\[ y = \frac {i \left ({\mathrm e}^{i t}-{\mathrm e}^{-i t}\right ) \ln \left (\frac {i {\mathrm e}^{i t}-1}{-{\mathrm e}^{i t}+i}\right )}{2}+c_1 \sin \left (t \right )-c_2 \cos \left (t \right )-\ln \left (i+{\mathrm e}^{i t}\right )-\ln \left ({\mathrm e}^{i t}-i\right )+c_3 +\ln \left ({\mathrm e}^{i t}\right ) \]

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 35

DSolve[D[y[t],{t,3}]+D[y[t],t]==Tan[t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\sin (t) \text {arctanh}(\sin (t))-\frac {1}{2} \log \left (\cos ^2(t)\right )-c_2 \cos (t)+c_1 \sin (t)+c_3 \]