Question

Determine whether each first-order differential equation is separable, linear, both, or neither. ---------------- 1. dy/dx+eˣy= x²y² ---------------- 2. y+eˣsin x = x³y' ---------------- 3 .In x-x²y=xy' --------------- 4. dy/dx + cos y= tan x

Answer 1

None of the provided differential equations are both linear and separable. Each equation either involves products of functions of y, non-linear terms, or functions that prevent separation of variables, eliminating them from being categorized as linear or separable.

Step-by-step explanation:

Classification of Differential Equations

To determine the type of each differential equation, whether it is separable, linear, both, or neither, we must analyze their structure:

1. dy/dx + eˣy = x²y²:
   This equation is non-linear because it contains a product of the functions of y (x²y²), and it is non-separable because you cannot separate the y terms and the x terms on opposite sides of the equation due to the product x²y².
2. y + eˣsin x = x³y':
   This equation is non-linear since y' is multiplied by a function of x (x³), making it a non-linear first-order differential equation. It is also non-separable.
3. ln x - x²y = xy':
   This equation is not linear because it contains the term x²y, and it can't be considered separable as the terms involving y and its derivative y' are not easily separated due to the presence of the x²y term.
4. dy/dx + cos y = tan x:
   This equation is neither linear nor separable. The presence of cosine and tangent functions involving both y and x complicates direct separation or conversion into a linear form.

In summary, none of the given differential equations are linear nor separable.