Internal problem ID [14688]
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: 13.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _missing_y]]
\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime }=\sec \left (2 t \right )^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 364
dsolve(diff(y(t),t$4)+4*diff(y(t),t$2)=sec(2*t)^2,y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\pi \,{\mathrm e}^{-2 i t} \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )}{64}+\frac {{\mathrm e}^{2 i t} \pi \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right )+1\right ) \operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1}{{\mathrm e}^{4 i t}+1}\right )}{64}+\frac {\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) {\mathrm e}^{-2 i t}}{64}+\frac {\pi \,\operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right ) {\mathrm e}^{2 i t}}{64}+\frac {i {\mathrm e}^{2 i t} \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )}{32}-\frac {i \ln \left ({\mathrm e}^{i t}\right ) t}{4}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) {\mathrm e}^{-2 i t}}{64}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) {\mathrm e}^{2 i t}}{64}+\frac {\left (-1+i {\mathrm e}^{-2 i t}\right ) \ln \left ({\mathrm e}^{-4 i t}+1\right )}{32}-\frac {i {\mathrm e}^{-2 i t} \ln \left (i \left ({\mathrm e}^{-2 i t}-i\right )^{2}\right )}{32}-\frac {{\mathrm e}^{-2 i t} \left (c_{2} i+c_{1} \right )}{8}+\frac {\left (-i {\mathrm e}^{2 i t}-1\right ) \ln \left ({\mathrm e}^{4 i t}+1\right )}{32}+\frac {\left (c_{2} i-c_{1} \right ) {\mathrm e}^{2 i t}}{8}-\frac {t^{2}}{4}+c_{3} t +c_{4} \]
✓ Solution by Mathematica
Time used: 0.256 (sec). Leaf size: 55
DSolve[y''''[t]+4*y''[t]==Sec[2*t]^2,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{32} \left (32 c_4 t-\log \left (\cos ^2(2 t)\right )-8 c_1 \cos (2 t)-2 \sin (2 t) \coth ^{-1}(\sin (2 t))-8 c_2 \sin (2 t)+32 c_3\right ) \]