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 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

\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}\)

It can be written as \(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}={x}{\left[{y}{x}^{{4}}-{1}\right]}\)

\(\displaystyle\frac{{1}}{{x}}{\left.{d}{y}\right.}={\left({y}{x}^{{4}}-{1}\right)}{\left.{d}{x}\right.}\)

The function of x, dx and y,dy can not be written on different sides, so the equation is not separable

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

Here \(\displaystyle{M}={y}{x}^{{4}}-{1}{\quad\text{and}\quad}{N}=\frac{{1}}{{x}}\)

Now, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={x}^{{4}}{\quad\text{and}\quad}{\frac{{\partial{n}}}{{\partial{x}}}}=-\frac{{1}}{{x}^{{2}}}\)

So, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}\ne{\frac{{\partial{n}}}{{\partial{x}}}}\). This is not 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

\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}\)

implies \(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}-{x}^{{5}}{y}=-{x}\) 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 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

\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}\)

It can be written as \(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}={x}{\left[{y}{x}^{{4}}-{1}\right]}\)

\(\displaystyle\frac{{1}}{{x}}{\left.{d}{y}\right.}={\left({y}{x}^{{4}}-{1}\right)}{\left.{d}{x}\right.}\)

The function of x, dx and y,dy can not be written on different sides, so the equation is not separable

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

Here \(\displaystyle{M}={y}{x}^{{4}}-{1}{\quad\text{and}\quad}{N}=\frac{{1}}{{x}}\)

Now, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}={x}^{{4}}{\quad\text{and}\quad}{\frac{{\partial{n}}}{{\partial{x}}}}=-\frac{{1}}{{x}^{{2}}}\)

So, \(\displaystyle{\frac{{\partial{M}}}{{\partial{y}}}}\ne{\frac{{\partial{n}}}{{\partial{x}}}}\). This is not 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

\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}\)

implies \(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}-{x}^{{5}}{y}=-{x}\) which is not the required form. So this is not a Bernoulli's equation.