Find if the following first order differential equations seperable (x)(e^y)(dy/dx) = e^(-2x) + e^(y-2x)

avissidep

avissidep

Answered question

2021-09-15

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.
(x)(ey)(dydx)=e2x+ey2x

Answer & Explanation

timbalemX

timbalemX

Skilled2021-09-16Added 108 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 N 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 (x)(ey)(dydx)=e2x+ey2x
It can be written as (x)(ey)(dydx)=e2x+ey2x
implies (x)(ey)(dydx)=e2xeye2x
implies (x)(ey)(dydx)=e2x(1ey)
implies ey1eydy=1xe2xdx
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 M=1xe2xandN=ey1ey
Now, My=0andnx=0
So, My=nx. This is an exact equation
The given equation can not be written in the form

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