74.14.28 problem 28

Internal problem ID [16433]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 28
Date solved : Tuesday, January 28, 2025 at 09:07:54 AM
CAS classification : [[_3rd_order, _missing_y]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.282 (sec). Leaf size: 63

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

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

✓ Solution by Mathematica

Time used: 60.029 (sec). Leaf size: 56

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

\[ y(t)\to \int _1^t(\cos (K[1]) \log (\cos (K[1]))+(K[1]+1) \sin (K[1]))dK[1]-\int _1^0(\cos (K[1]) \log (\cos (K[1]))+(K[1]+1) \sin (K[1]))dK[1] \]