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. −[y4−(t)(y3)]dt=(3)(t2)(y2)dy

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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.  −[y4−(t)(y3)]dt=(3)(t2)(y2)dy

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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∂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 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 −[y4−(t)(y3)]dt=(3)(t2)(y2)dy It can be written as (dydt)=−y4−(t)(y3)(3)(t2)(y2) The function of t, dt and y,dy can not be written on different sides, so the equation is notseparable It has products of dependent variables, so this is not a linear differential equation. Here M=−y4+ty3andN=3t2y2 Now, ∂M∂y=−y4+ty3 and delndelx=6ty2 So, ∂M∂y≠∂n∂x. This is not an exact equation The given equation can be written as −y4−(t)(y3)(3)(t2)(y2)= =−yt−13(ty), dividing by ty3 =−13(yt−1)(yt) So, it is a homogeneous equation.

Rewrite the equation into standard form for the classification it fits.-[y^4 - (t)(y^3)]dt = (3)(t^2)(y^2)dy

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