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I essentially want to get my equation to look like this, but can't quite get it. I've used drcases but then the alignment is difficult. I've also tried empheq, but still having issues. Any ideas?

You can use aligned inside dcases from the mathtools package. This is how the picture aligned.

documentclass[a4paper]{article}

usepackage{mathtools}



begin{document}



begin{equation}

begin{dcases}left.

begin{aligned}

frac{dG(t)}{dt} &= G_{in} - sigma_2G - aleft(c+frac{mI}{n+I}right)G + b \

frac{dI(t)}{dt} &= frac{sigma_1G^2}{alpha_1^2+G^2} - d_iI(t)

end{aligned} right }, quad t ne k_tau,  & \

left.

begin{aligned}

G(t^+) &= G(t) \

I(t^+) &= I(t) + sigma

end{aligned} right }, quad t = k_tau,  & 

end{dcases}

end{equation}



end{document}

Update:

You can also align all the equation at the = sign like this:

documentclass[a4paper]{article}

usepackage{mathtools,calc}



newlength{lhs}

settowidth{lhs}{ $frac{dG(t)}{dt}$ }

newcommand{lhsbox}[1]{makebox[lhs][r]{$displaystyle#1$}}



begin{document}



begin{equation}

begin{dcases}left.

begin{aligned}

lhsbox{frac{dG(t)}{dt}} &= G_{in} - sigma_2G - aleft(c+frac{mI}{n+I}right)G + b \

lhsbox{frac{dI(t)}{dt}} &= frac{sigma_1G^2}{alpha_1^2+G^2} - d_iI(t)

end{aligned} right }, quad t ne k_tau,  & \

left.

begin{aligned}

lhsbox{G(t^+)} &= G(t) \

lhsbox{I(t^+)} &= I(t) + sigma

end{aligned} right }, quad t = k_tau,  & 

end{dcases}

end{equation}



end{document}

The following uses eqparbox's eqmakebox[<tag>][<align>]{<stuff>} to ensure all <stuff> with the same <tag> is set in a box of maximum width with the appropriate <align>ment. This automates the process of aligning the equation left-hand sides:

documentclass{article}



usepackage{mathtools,eqparbox}



begin{document}



begin{equation}

  begin{dcases}

    left. begin{aligned}

      & eqmakebox[LHS][r]{$dfrac{mathrm{d}G(t)}{mathrm{d}t}$} 

        = G_{in} - sigma_2 G - a left(c + frac{mI}{n + I} right) G + b \

      & eqmakebox[LHS][r]{$dfrac{mathrm{d}I(t)}{mathrm{d}t}$} 

        = frac{sigma_1 G^2}{alpha_1^2 + G^2} - d_i I(t)

    end{aligned} right } quad t neq k_tau, \

    left. begin{aligned}

      & eqmakebox[LHS][r]{$G(t^+)$} = G(t) \

      & eqmakebox[LHS][r]{$I(t^+)$} = I(t) + sigma

    end{aligned} right } quad t = k_tau

  end{dcases}

end{equation}



end{document}

Here's a solution that employs nested array environments.

documentclass{article}

usepackage{array,booktabs}

newcolumntype{L}{>{displaystyle}l}

begin{document}

[

renewcommandarraystretch{1.33}

setlengtharraycolsep{0pt}

left{ 

begin{array}{L}

left.

  begin{array}{L}

    a = ldfjdl;kfsja ;kfja;ksdljfa\

    b = asjfl;a ;adjfl;asj ;adjfals

  end{array} 

right}

,quad tne kr,. \ addlinespace

left.

  begin{array}{L}

    c = jfdals;jkf\

    d = dsalfjasl;fja

  end{array} 

right}

,quad t=kr,.

end{array}

right.

]

end{document}