Question

Without solving, classify each of the following equations as to whether it is separable, homogenous, exact, linear, or Bernoulli (circle all applicable).

(a). dy/dx = 1/y-x
(b). .dy/dx = x-y/x
(c). dy/dx = 1/x(x-y)
(d). dy/dx = y²+y/x²+x

Answer 1

The classification of each equation without solving is provided. Equation (a) is separable, exact, linear, and Bernoulli. Equation (b) is linear. Equation (c) is separable and exact. Equation (d) is nonlinear.

Step-by-step explanation:

(a) The equation dy/dx = 1/y-x is separable, exact, linear, and Bernoulli. This equation can be separated into two integrals: ∫1/y dy = ∫1-x dx, making it separable. It is also exact because the partial derivative of ∫1/y dy with respect to y is equal to the partial derivative of ∫1-x dx with respect to x. Furthermore, it is linear because all the terms involving y and x have degree 1. Finally, it is Bernoulli because it can be transformed into a linear equation by multiplying both sides by y and rewriting it as y(dy/dx) - y/x = 1.

(b) The equation dy/dx = x-y/x is linear. It is linear because all the terms involving y and x have degree 1.

(c) The equation dy/dx = 1/x(x-y) is separable and exact. It can be separated into two integrals: ∫1/(x-y) dy = ∫1/x dx, making it separable. It is also exact because the partial derivative of ∫1/(x-y) dy with respect to y is equal to the partial derivative of ∫1/x dx with respect to x.

(d) The equation dy/dx = y²+y/x²+x is nonlinear. It is nonlinear because it contains terms involving y², y, x², and x to different degrees.