For each of the following ordinary differential equations determine if it is linear or nonlinear and determine its order.
\(\dot{y} + 5 y = t^2 + 2 \,, \quad\) where \(y\) is a function of \(t\).
\(\dfrac{\dx}{\dt} = x^2 \,, \quad\) where \(x\) is a function of \(t\).
\(x y^{(5)} + x^2 y^{(3)} + \sin(x) y' - y = e^x \,, \quad\) where \(y\) is a function of \(x\).
\(\left( \dfrac{\dy}{\dt} \right) \left( \dfrac{\dee^2 y}{\dt^2} \right) + y = 0 \,, \quad\) where \(y\) is a function of \(t\).
\(x'' + t^2 x' + 3 t = 0 \,, \quad\) where \(x\) is a function of \(t\).
\(u \, \dfrac{du}{dt} + t^2= 3u \,, \quad\) where \(u\) is a function of \(t\).
\(\sin(t) \, \dfrac{\dee^2 x}{\dt^2} + \cos(t) \, \dfrac{\dx}{\dt} = \tan(t) \,, \quad\) where \(x\) is a function of \(t\).
For each of the following systems of ordinary differential equations determine if it is linear or nonlinear and determine its order.
\[\begin{aligned} y' & = y + t x \,, \\ x' & = y^2 + 3 x \,, \end{aligned} \quad \text{where $x$ and $y$ are both functions of $t$} \,.\]
\[\begin{aligned} y'' & = y' + t x' \,, \\ x' & = y + 3 x \,, \end{aligned} \quad \text{where $x$ and $y$ are both functions of $t$} \,.\]
\[\begin{aligned} x' & = x + t^2 y + 6 z \,, \\ y' & = 3 x + y + e^t \,, \\ z' & = y + t z \,, \end{aligned} \quad \text{where $x,y$ and $z$ are functions of $t$} \,.\]
For each of the following partial differential equations determine if it is linear or nonlinear and determine its order.
\(\del_{xx} \phi + \del_{yy} \phi = 0 \,, \quad\) where \(\phi\) is a function of \(x\) and \(y\).
\(-\del_y \phi = (\del_x \phi)^2/4 \, \quad\) where \(\phi\) is a function of \(x\) and \(y\).