Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve. a) dy/dx = (x − y)/x
b) (x + 1)dy/dx = −y + 20
c) dy/dx = 1/(x(x − y2))
d) dy/dx =(y^2 + y)/(x^2 + x)
e) dy/dx = 5y + y^2
f) y dx = (y − xy^2) dy
g) x dy/dx = ye^(xy) − x
h) 2xyy' + y^2 = 2x^2
i) y dx + x dy = 0
k) (x^2 + 2y/x) dx = (3 − ln x^2) dy
l) (y/x^2) dy/dx + e^(2x^3) + y^2 = 0

a) dy/dx = (x − y)/x. This is exact, linear in y, and homogeneous.

b) (x + 1)dy/dx = −y + 20. This is separable, exact, and liner in x and y.

c) dy/dx = 1/(x(x − y2)). This is Bernoulli in x.

d) dy/dx =(y^2 + y)/(x^2 + x). This is Bernoulli in x and y, and Separable.

e) dy/dx = 5y + y^2. This is Bernoulli in y, and Separable.

f) y dx = (y − xy^2) dy. This is linear in x.

g) x dy/dx = ye^(xy) – x. This is homogeneous.

h) 2xyy' + y^2 = 2x^2. This is exact, homogeneous, and Bernoulli.

i) y dx + x dy = 0. This is exact, homogeneous, separable, and linear in x and y.

k) (x^2 + 2y/x) dx = (3 − ln x^2) dy. This is exact, and linear in y.

l) (y/x^2) dy/dx + e^(2x^3) + y^2 = 0. This is separable.

Step-by-step explanation:

The following categories of each of the differential equation are first explained as follows:

Separable differential equation: Any equation that can be expressed in the form y′=f(x)g is a separable differential equation (y).

Exact differential equation: This is a type of differential equation that can be solved directly without the use of any of the special techniques in the subject.

Linear differential equation: This is a form of differential equation that can be solved without resorting to any of the subject's unique techniques.

Homogeneous differential equation: A differential equation is said to be homogeneous if it is a homogeneous function of the unknown function and its derivatives.

Bernoulli differential equation: If an ordinary differential equation has the form y' + P(x)y = Q(x)y^n, where n is a real number, it is referred to as a Bernoulli differential equation.

Since the question indicates "Do not solve", we therefore only classify as follows:

a) dy/dx = (x − y)/x. This is exact, linear in y, and homogeneous.

b) (x + 1)dy/dx = −y + 20. This is separable, exact, and liner in x and y.

c) dy/dx = 1/(x(x − y2)). This is Bernoulli in x.

d) dy/dx =(y^2 + y)/(x^2 + x). This is Bernoulli in x and y, and Separable.

e) dy/dx = 5y + y^2. This is Bernoulli in y, and Separable.

f) y dx = (y − xy^2) dy. This is linear in x.

g) x dy/dx = ye^(xy) – x. This is homogeneous.

h) 2xyy' + y^2 = 2x^2. This is exact, homogeneous, and Bernoulli.

i) y dx + x dy = 0. This is exact, homogeneous, separable, and linear in x and y.

k) (x^2 + 2y/x) dx = (3 − ln x^2) dy. This is exact, and linear in y.

l) (y/x^2) dy/dx + e^(2x^3) + y^2 = 0. This is separable.

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