Vtyjkyuk

I try with some recommendations to add vertical space for cells with equations, and I couldn't achieve it.

Here is my MWE:

documentclass[12pt,a4paper]article
 usepackagearray
usepackagemakecell
renewcommandtheadfontnormalsizebfseries
usepackage[math]cellspace
setlengthcellspacetoplimit4pt
setlengthcellspacebottomlimit4pt

usepackagecolor,colortbl,hhline
definecolorGraygray0.9
usepackagecaption
usepackageamsmath

begindocument 
begintable[htpb]
 centering
 captionSolución de la ecuación 
 begintabularp12cm
 hline
 rowcolorGray theadCaso & theadSolución \
 hline
 $zeta = 1$ & $u(t) = [u(0) + (dot u(0) + u(0) omega_n )t]e^-omega_n t$\hline
 $zeta > 1$ & $u(t) = dfrace^-omega_nzeta2omega_nsqrtzeta^2-1leftlbraceleft[omega_n(zeta+sqrtzeta^2-1)u(0)+dot u(0)right]e^omega_nsqrtzeta^2-1-big[omega_n(zeta - sqrtzeta^2-1)u(0)+dot u(0)big]e^-omega_nsqrtzeta^2-1rightrbrace$ \hline
 $zeta < 1$ & $u(t) = e^-omega_nzeta tleft[u(0)cosomega_Dt + left(dfracdot u(0) + omega_nzeta u(0)omega_Dright)right]$ \hline
 endtabular
 endtable
enddocument

Update

Maybe I can't anchieve the solution because of conflicts between some packages. This is my preamble, and the Typeset Engine is pdfLaTeX:

documentclass[12pt,a4paper]article

%usepackagepolyglossia
%setdefaultlanguage[babelshorthands]spanish
%usepackage[Ligatures=TeX]fontspec
usepackage[spanish,es-noshorthands]babel
usepackage[T1]fontenc%http://tex.stackexchange.com/questions/44694/fontenc-vs-inputenc
usepackage[utf8]inputenc
usepackagemathptmx%http://tex.stackexchange.com/questions/669/how-do-i-make-my-document-use-the-times-font-both-for-the-text-and-the-math
%fontspecTimes New Roman
%usepackagefontspec
%setmainfontTimes New Roman
%usepackage[T1]fontenc
%usepackageifthen
%usepackagefontspec
%setmainfontArial

usepackage[table]xcolor

%usepackage[htt]hyphenat
%
%usepackagepdflscape

%usepackage[title]appendix

usepackage[style=apa]biblatex
addbibresourcebibltarea6.bib

%usepackagearray,multirow
usepackagemakecell
renewcommandtheadfontnormalsizebfseries
renewcommandtheadgape
usepackage[math]cellspace
setlengthcellspacetoplimit4pt
setlengthcellspacebottomlimit4pt

usepackagelistings
lstset
basicstyle = ttfamily,
keywordstyle= colorbluetextbf,
commentstyle= colorgray,
stringstyle= colorgreen!70!black,
stringstyle= colorred,
columns= fullflexible,
numbers= left,
numberstyle= scriptsizesffamilycolorgray,
xleftmargin= 0.07textwidth,
xrightmargin= 0.05textwidth,
showstringspaces = false,


%usepackageinconsolata

%newcommandCppCtexttt++
%renewcommandlstlistingnameCódigo
%renewcommandlstlistlistingnameÍndice de códigos

%usepackagetikz
%usepgflibraryarrows.meta
%tikzset%
% newTip/.tip=Bar[sep=-1.45pt, width=10pt] Circle[sep=-1.25pt,width=2.5pt,length=2.5pt]
% 
%usetikzlibrarypatterns
%
%usepackagetikz
%usepgflibraryarrows.meta,shapes.arrows
%tikzset%
% newTip/.tip=Bar[sep=-1.45pt, width=10pt] Circle[sep=-1.25pt,width=2.5pt,length=2.5pt]
% 
%usetikzlibrarypatterns
%usetikzlibrarycalc

%usepackagecaption
%usepackagesubcaption

usepackageamsmath
usepackagemathtools
%setcounterMaxMatrixCols25

usepackagesiunitx
sisetupper-mode = symbol

%usepackagepgfplots

%newcountermyrow

%usepackagelongtable
%usepackagethreeparttable

%usepackage[inline]enumitem
%usepackagehyperref

usepackagetitlesec
titleformat*sectionbfseriesnormalsize
titleformat*subsectionbfseriesnormalsize
titleformat*subsubsectionbfseriesnormalsize

usepackage[ampersand]easylist

%usepackagelayout

usepackagecolor,colortbl,hhline
definecolorGraygray0.9

%usepackage[a4paper,left=3cm, top=3cm, right=2.5cm, bottom=2.5cm]geometry

setlengthhoffset.46cm
setlengthoddsidemargin13.08pt
setlengthevensidemargin0cm
setlengthmarginparwidth0cm
setlengthtopmargin-7mm
setlengthmarginparsep0cm
setlengthheadheight8mm
setlengthheadsep4.6mm
setlengthtextheight24cm
setlengthfootskip8mm
setlengthtextwidth15.5cm

usepackagefancyhdr
pagestylefancy
 lheadfootnotesizePontificia Universidad Católica del Perú\ footnotesizeEscuela de Posgrado - Ingeniería Civil%Significa que est
 chead%Vac
 rheadfootnotesizeDinámica de Estructuras\ Tarea 1
 lfootfootnotesize%Pie izquierdo
 cfootthepage
 rfootfootnotesizeFernández Yopla, Isai

 renewcommandheadrulewidth0.4pt%Falta investigar
 renewcommandfootrulewidth0.4pt

usepackage[pass]geometry%pass tells output driver which paper size use.

equation in the second row/column is far to long. you need to split it in two lines. for example with use the aligned environment from the package mathtools. for more vertical space you should used S column specifier from cellspace. for example for the second column: Sp12cm:

edit:

• in cases, when the siunitx package is used, the S specifier from the cellspace package had to be replaced by C, i.e. Cp(12cm)
  


• in mwe below i consider only for table relevant packages


• off-topic: since you also use the geometry package, you can define page layout with it and remove all other low level page layout definitions in your preamble

documentclass[12pt,a4paper]article
usepackage[T1]fontenc%http://tex.stackexchange.com/questions/44694/fontenc-vs-inputenc
usepackage[utf8]inputenc
usepackage[spanish,es-noshorthands]babel
usepackagemathptmx%http://tex.stackexchange.com/questions/669/how-do-i-make-my-document-use-the-times-font-both-for-the-text-and-the-math

usepackagemakecell,hhline % <---
renewcommandtheadfontnormalsizebfseries
renewcommandtheadgape
usepackagecellspace % <---
setlengthcellspacetoplimit4pt
setlengthcellspacebottomlimit4pt

usepackagemathtools % it load "amsmath" too
usepackagesiunitx % <--- used it, the "S" from the "cellspace# package
 % had to be replaced with "C"
usepackage[table]xcolor % <---
definecolorGraygray0.9
usepackagegeometry % for simple determination of page layout.
 % see its documentation!

begindocument
 begintable[htpb]
 centering
captionSolución de la ecuación 
 begintabularCp12cm
 hline
 rowcolorGray
 theadCaso & theadSolución \
 hline
 $zeta = 1$ & $u(t) = bigl[u(0) + bigl(dotu(0) + u(0) omega_nbigr) tbigr] e^-omega_n t$ \ hline
 $zeta > 1$ & $beginmultlined
 u(t) = dfrace^-omega_nzeta2omega_nsqrtzeta^2-1
 bigglBigl[omega_nBigl(zeta+sqrtzeta^2-1Bigr)u(0)+dotu(0)Bigr] e^omega_nsqrtzeta^2-1 - \
 Bigl[omega_nBigl(zeta - sqrtzeta^2-1Bigr)u(0)+dotu(0)Bigr] e^-omega_nsqrtzeta^2-1
 biggr
 endmultlined$ \ hline
 $zeta < 1$ & $u(t) = e^-omega_nzeta t
 biggl[u(0)cosomega_Dt + 
 biggl(dfracdotu(0) + omega_nzeta u(0)omega_Dbiggr)
 biggr]$ \ hline
 endtabular
 endtable
enddocument

As @Zarko has already pointed out in his answer, the main problem is that one of the main equations is far too long to fit in a single row. Use an aligned environment to introduce a suitably chosen line break.

Once you've line-broken the long equation, there's no need to use p column types; instead, just use the basic l column type for both columns.

In order to declutter the tabular material some more, I would remove the repetitive u(t) = parts from the cells and, instead, stick that information into the table's caption.

I would also like to suggest that you give the table a much more open "look", mainly by getting rid of all vertical and horizontal black lines -- they're simply not needed. I'd also load the booktabs package and use its addlinespace macro to further increase whitespace between rows.

In addition, I would replace all left and right directives with explicit sizing instructions.

documentclass[12pt,a4paper]article
usepackage[spanish]babel % new
usepackagearray
usepackagebooktabs % for "addlinespace" macro
usepackage[table]xcolor
definecolorGraygray0.9
usepackage[skip=0.333baselineskip]caption
usepackageamsmath % for 'aligned' environment

begindocument 
begintable[htpb]
setlengthextrarowheight3pt
centering
captionSolución de la ecuación por $u(t)$
begintabularll
rowcolorGray textbfCaso & textbfSolución \
addlinespace
$zeta = 1$ & 
$displaystyle
 bigl[u(0) + (dot u(0) + u(0) omega_n )tbigr]
 e^-omega_n t
$\ addlinespace
$zeta > 1$ & 
$beginaligned[t]
frace^-omega_nzeta2omega_nsqrtzeta^2-1Bigllbrace
 &bigr[omega_n(zeta+sqrtzeta^2-1)u(0)+dot u(0)bigr] 
 e^omega_nsqrtzeta^2-1\
-&bigl[omega_n(zeta - sqrtzeta^2-1)u(0)+dot u(0)bigr]
 e^-omega_nsqrtzeta^2-1,
Bigrrbrace
endaligned$ \ addlinespace
$zeta < 1$ & 
$displaystyle 
e^-omega_nzeta tbigl[u(0)cosomega_Dt 
+ bigl(dot u(0) + omega_nzeta u(0)bigr)big/omega_D
bigr]$ \
endtabular
endtable
enddocument