Separation of variables: If in an equation, it is possible to get all the functions of x and dx to one side and all the functions of y and dy to the other, the variables are said to be separable.

Linear Differential Equation: A differential equation is called linear if every dependent variable and every derivative involved occurs in the first degree only, and no products of dependent variables and/or derivatives occur.

Exact Differential Equation: The necessary and efficient for the differential equation \(\displaystyle{\left({M}\right)}{\left.{d}{x}\right.}+{\left({N}\right)}{\left.{d}{y}\right.}={0}\) to be exact is

\(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={\frac{{\partial{n}}}{{\partial{x}}}}\)

Homogeneous equation: A differential equation of first order and first N degree is said to be homogeneous if it can be put in the form

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={f{{\left({\frac{{{y}}}{{{x}}}}\right)}}}\)

Or, equations of the type \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{A}{\left({x},{y}\right)}}}{{{B}{\left({x},{y}\right)}}}}\) where

\(\displaystyle{A}{\left(\lambda{x},\lambda{y}\right)}=\lambda^{{d}}{A}{\left({x},{y}\right)}\)

\(\displaystyle{B}{\left(\lambda{x},\lambda{y}\right)}=\lambda^{{d}}{B}{\left({x},{y}\right)}\)

are homogeneous equations of degree d.

Bernoulli’s Equation: An equation of the form

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{P}{y}={Q}{y}^{{n}}\)

where P and Q are constants or functions of x alone (amd not of y) and n is constant except 0 and 1, is called a Bernoulli’s differential equation.

The given differential equation is \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{{y}-{2}{x}}}\)

It can be written as \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{{y}-{2}{x}}}\)

implies \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}-{e}^{{y}}{e}^{{-{2}{x}}}\)

implies \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}{\left({1}-{e}^{{y}}\right)}\)

implies \(\displaystyle{\frac{{{e}^{{y}}}}{{{1}-{e}^{{y}}}}}{\left.{d}{y}\right.}=\frac{{1}}{{x}}{e}^{{-{2}{x}}}{\left.{d}{x}\right.}\)

The function of x, dx and y,dy can be written on different sides, so the equation is separable

It has products of dependent variables, so this is not a linear differential equation.

Here \(\displaystyle{M}=\frac{{1}}{{x}}{e}^{{-{2}{x}}}{\quad\text{and}\quad}{N}={\frac{{{e}^{{y}}}}{{{1}-{e}^{{y}}}}}\)

Now, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={0}{\quad\text{and}\quad}{\frac{{\partial{n}}}{{\partial{x}}}}={0}\)

So, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={\frac{{\partial{n}}}{{\partial{x}}}}\). This is an exact equation

The given equation can not be written in the form \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={f{{\left({\frac{{{y}}}{{{x}}}}\right)}}}.\)

So, it is not a homogeneous equation

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

implies \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{y}}{e}^{{{2}{x}}}\)

implies \(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}-{\frac{{{1}}}{{{x}}}}{e}^{{-{y}}}{e}^{{-{2}{x}}}=-\frac{{1}}{{x}}{e}^{{-{2}{x}}}\)

which is not the required form. So this is not a Bernoulli's equation.

Linear Differential Equation: A differential equation is called linear if every dependent variable and every derivative involved occurs in the first degree only, and no products of dependent variables and/or derivatives occur.

Exact Differential Equation: The necessary and efficient for the differential equation \(\displaystyle{\left({M}\right)}{\left.{d}{x}\right.}+{\left({N}\right)}{\left.{d}{y}\right.}={0}\) to be exact is

\(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={\frac{{\partial{n}}}{{\partial{x}}}}\)

Homogeneous equation: A differential equation of first order and first N degree is said to be homogeneous if it can be put in the form

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={f{{\left({\frac{{{y}}}{{{x}}}}\right)}}}\)

Or, equations of the type \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{A}{\left({x},{y}\right)}}}{{{B}{\left({x},{y}\right)}}}}\) where

\(\displaystyle{A}{\left(\lambda{x},\lambda{y}\right)}=\lambda^{{d}}{A}{\left({x},{y}\right)}\)

\(\displaystyle{B}{\left(\lambda{x},\lambda{y}\right)}=\lambda^{{d}}{B}{\left({x},{y}\right)}\)

are homogeneous equations of degree d.

Bernoulli’s Equation: An equation of the form

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{P}{y}={Q}{y}^{{n}}\)

where P and Q are constants or functions of x alone (amd not of y) and n is constant except 0 and 1, is called a Bernoulli’s differential equation.

The given differential equation is \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{{y}-{2}{x}}}\)

It can be written as \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{{y}-{2}{x}}}\)

implies \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}-{e}^{{y}}{e}^{{-{2}{x}}}\)

implies \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}{\left({1}-{e}^{{y}}\right)}\)

implies \(\displaystyle{\frac{{{e}^{{y}}}}{{{1}-{e}^{{y}}}}}{\left.{d}{y}\right.}=\frac{{1}}{{x}}{e}^{{-{2}{x}}}{\left.{d}{x}\right.}\)

The function of x, dx and y,dy can be written on different sides, so the equation is separable

It has products of dependent variables, so this is not a linear differential equation.

Here \(\displaystyle{M}=\frac{{1}}{{x}}{e}^{{-{2}{x}}}{\quad\text{and}\quad}{N}={\frac{{{e}^{{y}}}}{{{1}-{e}^{{y}}}}}\)

Now, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={0}{\quad\text{and}\quad}{\frac{{\partial{n}}}{{\partial{x}}}}={0}\)

So, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={\frac{{\partial{n}}}{{\partial{x}}}}\). This is an exact equation

The given equation can not be written in the form \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={f{{\left({\frac{{{y}}}{{{x}}}}\right)}}}.\)

So, it is not a homogeneous equation

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

implies \(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{y}}{e}^{{{2}{x}}}\)

implies \(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}-{\frac{{{1}}}{{{x}}}}{e}^{{-{y}}}{e}^{{-{2}{x}}}=-\frac{{1}}{{x}}{e}^{{-{2}{x}}}\)

which is not the required form. So this is not a Bernoulli's equation.