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)]

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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)]

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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∂M∂y=∂n∂xHomogeneous equation: A differential equation of first order and first degree is said to be homogeneous if it can be put in the formdydx=f(yx)Or, equations of the type  dydx=A(x,y)B(x,y) whereA(λx,λy)=λdA(x,y)B(λx,λy)=λdB(x,y)are homogeneous equations of degree d.Bernoulli’s Equation: An equation of the formdydx+Py=Qynwhere 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[yx4−1]1xdy=(yx4−1)dxThe function of x, dx and y,dy can not be written on different sides, so the equation is not separableIt has products of dependent variables, so this is not a linear differential equation. Here M=yx4−1andN=1xNow, ∂M∂y=x4and∂n∂x=−1x2So, ∂M∂y≠∂n∂x. 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&#039;s equation.

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)]

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