Step 1

For First order:

On simultaneous ordinary differential equation of the first order in which there are one or more than one variable in the number of equations.

Examples:

y'=y+2x

\(\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={3}{y}+{2}{x}\)

Step 2

For Second order:

The simultaneous ordinary differential equations of the second order has one or more than one variable in the number of equations.

Examples:

\(\displaystyle{y}{''}+{4}{y}'+{4}{y}={25}{x}+{16}{e}^{{t}}\)

\(\displaystyle\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}+\frac{{5}}{{2}}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}+{9}{y}={88}{x}+{5}{e}^{{t}}\)

For First order:

On simultaneous ordinary differential equation of the first order in which there are one or more than one variable in the number of equations.

Examples:

y'=y+2x

\(\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={3}{y}+{2}{x}\)

Step 2

For Second order:

The simultaneous ordinary differential equations of the second order has one or more than one variable in the number of equations.

Examples:

\(\displaystyle{y}{''}+{4}{y}'+{4}{y}={25}{x}+{16}{e}^{{t}}\)

\(\displaystyle\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}+\frac{{5}}{{2}}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}+{9}{y}={88}{x}+{5}{e}^{{t}}\)