Question

Ordinary Differential Equation. Rewrite in the form of a first-order separable ODE and then isolate y:

A. dy​/dx=e^−y(2x−4)
B. y=−2ln(e^y−2x)
C. e^y=−x^2/2​+2x
D. y=−2x+4e^y

Answer 1

To rewrite the differential equation in the form of a first-order separable ODE and isolate y, follow specific steps for each given equation. A, B, and C can be rewritten in the form of a first-order separable ODE, while D cannot.

Step-by-step explanation:

To rewrite the differential equation in the form of a first-order separable ODE and isolate y:

A. Start by separating the variables by moving all terms involving y to one side and all terms involving x to the other side: dy/dx + e^y(2x - 4) = 0. Then, divide both sides by e^y and separate the dx and dy terms: e^(-y)dy = -(2x - 4)dx. This is the first-order separable ODE in the form of dy/dx = f(x)g(y).

B. The given equation does not need to be rewritten, as it is already in the form of a first-order separable ODE. The variable y is isolated on one side of the equation.

C. Start by moving the e^y term to the other side: e^y = -x^2/2 + 2x. Take the natural logarithm of both sides: ln(e^y) = ln(-x^2/2 + 2x). Simplify to get y = ln(-x^2/2 + 2x), which is in the form of a first-order separable ODE.

D. Start by moving the -2x term to the other side: dy/dx + 2x = -4e^y. This is not in the form of a first-order separable ODE, as y is not separated on one side of the equation.