(a) identify the independent variable and the dependent variable of each equation (use 't' for the independent variable if an independent variable is not given explicitly): (b) give the order of each differential equation (enter '1' for first order. '2' for second order and so on: do not include the quotes); and (c) state whether the equation is linear or nonlinear if your answer to (c) is nonlinear, make sure that you can explain why this is true

You didn't give the differential equations, but I'll explain how to identify the independent variable, dependent variable, how to know the order, linearity, and nonlinearity of a differential equation.

Step-by-step explanation:

DIFFERENTIAL EQUATION

This is any equation that involves differential coefficients. It is a relationship between an independent variable, x, a dependent variable, y, and one or more derivatives of y with respect to x.

Examples

(1) xd²y/dx² + 7dy/dx = 0

(2) y²dy/dx + 2x = 0

(3) xd³y/dx³ = y½ + 1

(4) 2xy'' - 3y' + 5y = 0

(5) (y''')² + 30xy = 0

Note that the dependent variable is always the numeratior, and the independent, denominator, in a different coefficient. In the case of our examples, y is the dependent variable, and x is the independent.

Example (4) is another way of writing a differential coefficient, y' (read as y-prime) is the same as dy/dx (read as dee-y dee-x). In some cases when the independent variable is time t, it is written as ÿ, which is the same as d²y/dt² (read as dee-two-y dee-t-squared)

ORDER

This is the order of the highest derivative in a differential equation. You need not consider other derivatives, just the highest.

In the examples, the orders are

(1) two

(2) one

(3) three

(4) two

(5) three

LINEAR DIFFERENTIAL EQUATION

This is the kind of differential equation in which the functions of the dependent variable are linear. There are no powers of the dependent variable and/or its derivatives, there are no products of the dependent variable and its derivative, there are no functions of the dependent variable like cos, sin, exp, etc.

NONLINEAR DIFFERENTIAL EQUATION

If any condition for linearity is not met, then it is nonlinear.

(1) Linear

(2) Nonlinear because y is the dependent variable, and y² is nonlinear, and even still, it multiplies a derivative.

(3) Nonlinear because y½ is nonlinear

(4) Linear

(5) Nonlinear because (y''')² in nonlinear.

Understanding this, you can determine the order, linearity or nonlinearity of any differential equation. Cheers!

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step-by-step explanation: