Question

Classify the following equation as exact, linear, or separable or none of the above.

a) dy/dx = (x − y)/x
b) (x + 1)dy/dx = −y + 20
c) dy/dx = 1/(x(x − y²))
d) dy/dx =(y² + y)/(x² + x)
e) dy/dx = 5y + y²

Answer 1

The classification of the differential equations is: a) separable, b) linear, c) none of the above, d) none of the above, and e) none of the above.

Step-by-step explanation:

When analyzing differential equations, we can classify them into different types based on their form. For each given equation, we will determine if it is exact, linear, separable, or none of the above.

• a) dy/dx = (x − y)/x: This is a separable equation because it can be written in the form of g(y)dy = f(x)dx by rearranging it to dy/(y − x) = dx/x.
• b) (x + 1)dy/dx = −y + 2: This is a linear equation since it can be expressed in the form dy/dx + P(x)y = Q(x), where P(x) = − 1/(x + 1) and Q(x) = 2/(x + 1).
• c) dy/dx = 1/(x(x − y²)): This equation is not exact, linear, or separable, so it is none of the above.
• d) dy/dx = (y² + y)/(x² + x): This equation is not exact, linear, or separable, so it is none of the above.
• e) dy/dx = 5y + y²: This equation does not fit the criteria for exact, linear, or separable equations, so it is also none of the above.