Classify the following differential equations as linear or nonlinear:

3x' +5x=7t^2
4x^2+2x+5tx=0

Evidently you have not reviewed the definition of a linear DE:

A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise.

So, the first above is linear, and the second is not.

To classify the given differential equations as linear or nonlinear, we need to examine their form.

1. 3x' + 5x = 7t^2
To determine if this equation is linear or nonlinear, we can check if the terms involving the dependent variable (in this case, x and its derivatives) are raised to powers other than 1 or multiplied together.

In this equation, we see that x' (the derivative of x with respect to t) and x are only raised to the first power and not multiplied together. Therefore, we can conclude that this differential equation is linear.

2. 4x^2 + 2x + 5tx = 0
Similarly, for this equation, we examine the form to determine if it is linear or nonlinear. We check if x and its derivatives are raised to powers other than 1 or multiplied together.

In this equation, we have the square term 4x^2, which indicates a nonlinear function. Therefore, we can classify this differential equation as nonlinear.

Summary:
1. 3x' + 5x = 7t^2 - Linear differential equation
2. 4x^2 + 2x + 5tx = 0 - Nonlinear differential equation