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

Kaycee Roche

Answered question

2021-09-13

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.
(dydx)=x2[(x3)(y)(1x)]

Answer & Explanation

Layton

Layton

Skilled2021-09-14Added 89 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 (M)dx+(N)dy = 0 to be exact is
My=nx
Homogeneous equation: A differential equation of first order and first degree is said to be homogeneous if it can be put in the form
dydx=f(yx)
Or, equations of the type dydx=A(x,y)B(x,y) where
A(λx,λy)=λdA(x,y)
B(λx,λy)=λdB(x,y)
are homogeneous equations of degree d.
Bernoulli’s Equation: An equation of the form
dydx+Py=Qyn
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
(dydx)=x2[(x3)(y)(1x)]
It can be written as (dydx)=x2[(x3)(y)(1x)]=x[yx41]
1xdy=(yx41)dx
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 M=yx41andN=1x
Now, My=x4andnx=1x2
So, Mynx. This is not an exact equation
The given equation can not be written in the form dydx=f(yx).
So, it is not a homogeneous equation
(dydx)=x2[(x3)(y)(1x)]
implies (dydx)x5y=x which is not the required form. So this is not a Bernoulli's equation.

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