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

Harlen Pritchard

Harlen Pritchard

Answered question

2021-09-03

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

Answer & Explanation

ottcomn

ottcomn

Skilled2021-09-04Added 97 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 [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, My=y4+ty3 and delndelx=6ty2
So, Mynx. This is not an exact equation
The given equation can be written as y4(t)(y3)(3)(t2)(y2)=
=yt13(ty), dividing by ty3
=13(yt1)(yt)
So, it is a homogeneous equation.

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