Determine whether each first-order differential equation is separable, linear, both, or neither.

dy/dx + e^x y = x^2 y^2

y + sin x = x^3 y'

In x - x^2 y = xy'

dy/dx + cos y = tan x

1. . It is not a first-order linear differential equation. And it's not separable either.

2. . It is a first-order linear differential equation.

3. . It is a first-order linear differential equation.

4. . It is not a first-order linear differential equation. And it's not separable either.

Step-by-step explanation:

Definition 1: A first-order linear differential equation is one that can be put into the form

where P and Q are continuous functions on a given interval.

Definition 2: A first-order differential equation is said to be separable if, after solving it for the derivative,

,

the right-hand side can then be factored as “a formula of just x” times “a formula of just y”,

If this factoring is not possible, the equation is not separable.

Applying the above definitions, we get that

1. For

It is not a first-order linear differential equation. And it's not separable either.

2. For

It is a first-order linear differential equation.

3. For

It is a first-order linear differential equation.

4. For

It is not a first-order linear differential equation. And it's not separable either.

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step-by-step explanation: