Answer:
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.
