\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

Classify the following differential equations, i.e., state their
order and degree. If one is first order, first degree, identify
their type. Do NOT attempt to solve them.

\begin{description}
\item[(i)]
$\ds\frac{dx}{dt}=kx$

\item[(ii)]
$\ds\frac{d^2y}{dx^2}+5\ds\frac{dy}{dx}+6y=10\sin x$

\item[(iii)]
$\left(\ds\frac{dy}{dx}\right)^2=x+y$

\item[(iv)]
$\ds\frac{d^3y}{dt^3}+\left(\ds\frac{dy}{dt}\right)^2=e^x$

\item[(v)]
$\cos x\left(\ds\frac{dy}{dx}\right)^6+\sin
x\left(\ds\frac{d^2y}{dx^2}\right)^3=0$

\item[(vi)]
$\left(\ds\frac{d^5x}{dt^5}\right)^2=x$
\end{description}

\medskip

{\bf Answer}
\begin{description}
\item[(i)]
${}$

$\begin{array} {rcl} \ds\frac{dx}{dt}\ \rm{is\ the\ highest\
derivative} & \Rightarrow & \rm{order}=1\\ \rm{power\ of}
\ds\frac{dx}{dt} \rm{is}\ 1 & \Rightarrow & \rm{degree}=1
\end{array}$

$\ds\frac{dx}{dt}=kx$ is un{variable} \un{separable}.

\item[(ii)]
${}$

$\begin{array} {rcl} \ds\frac{d^2y}{dx^2}\ \rm{is\ the\ highest\
derivative} & \Rightarrow & \rm{order}=2\\ \rm{power\ of}
\ds\frac{d^2t}{dx^2} \rm{is}\ 1 & \Rightarrow & \rm{degree}=1
\end{array}$

\item[(iii)]
${}$

$\begin{array} {rcl} \ds\frac{dy}{dx}\ \rm{is\ the\ highest\
derivative} & \Rightarrow & \rm{order}=1\\ \rm{power\ of}
\ds\frac{dy}{dx} \rm{is}\ 2 & \Rightarrow & \rm{degree}=1
\end{array}$

\item[(iv)]
${}$

$\begin{array} {rcl} \ds\frac{d^3y}{dt^3}\ \rm{is\ the\ highest\
derivative} & \Rightarrow & \rm{order}=3\\ \rm{power\ of}
\ds\frac{d^3y}{dt^3} \rm{is}\ 1 & \Rightarrow & \rm{degree}=1
\end{array}$

\item[(v)]
${}$

$\begin{array} {rcl} \ds\frac{d^2y}{dx^2}\ \rm{is\ the\ highest\
derivative} & \Rightarrow & \rm{order}=2\\ \rm{power\ of}
\ds\frac{d^2y}{dx^2} \rm{is}\ 3 & \Rightarrow & \rm{degree}=3
\end{array}$

\item[(vi)]
${}$

$\begin{array} {rcl} \ds\frac{d^5x}{dt^5}\ \rm{is\ the\ highest\
derivative} & \Rightarrow & \rm{order}=5\\ \rm{power\ of}
\ds\frac{d^5x}{dt^5} \rm{is}\ 2 & \Rightarrow & \rm{degree}=2
\end{array}$
\end{description}
\end{document}