Find if the following first order differential equations seperable, linear,exact,almost exact,homogeneous,or Bernoulli.(dy/dx) = x^2[(x^3)(y) - (1/x)]

Kaycee Roche 2021-09-13 Answered
Find if the following first order differential equations seperable, linear, exact, almost exact, homogeneous, or Bernoulli. Rewrite the equation into standard form for the classification it fits.
\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}\)

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Expert Answer

Layton
Answered 2021-09-14 Author has 8199 answers
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.
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