Internal problem ID [14702]
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: 27.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _missing_y]]
\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime }=\sec \left (t \right )^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.5 (sec). Leaf size: 71
dsolve([diff(y(t),t$4)+diff(y(t),t$2)=sec(t)^2,y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (\int _{0}^{t}\left (\left (-{\mathrm e}^{i \textit {\_z1}}-{\mathrm e}^{-i \textit {\_z1}}\right ) \ln \left (\frac {i {\mathrm e}^{i \textit {\_z1}}-1}{-{\mathrm e}^{i \textit {\_z1}}+i}\right )-2 i \ln \left ({\mathrm e}^{i \textit {\_z1}}\right )-2 \textit {\_z1} +2 \sin \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right )}{2}+t \]
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 28
DSolve[{y''''[t]+y''[t]==Sec[t]^2,{y[0]==0,y'[0]==1,y''[0]==0,y'''[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -2 \sin (t) \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )+t-\cos (t)-\log (\cos (t))+1 \]