A. Identify the independent variable and the dependent variable of each equation. b. Give the order of each differential equation.
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?

a. y'= y- x^2
b. xy'= 2y
c. x'' +5x =e^-x

Step-by-step explanation:

From the information given:

y' = y - x²

y' - y = -x²  

This is a first-order linear differential equation of the form y' + p(x)y = q(x)

Thus, y is dependent and x is said to be the independent variable.

xy' = 2y

This is also a first-order linear differential equation with y as the dependent variable and x as the independent variable.

x'' + 5x = e⁻ˣ

This is a second order non-linear differentiall equation.

Here, x is the dependent variable, and since the independent variable is not given, we will take t as the independent variable.

Equation       independent     dependent      order           linear/non linear

                     variable             variable    

y' = y - x²           x                         y                      1                  linear

xy' = 2y              x                         y                     1                   linear

x'' + 5x - e⁻ˣ       t                          x                      2                non-linear

Why this is true for (c) i.e. being non-linear is that; in the above two first-order linear differential equations, we will notice that the dependent variables and their derivatives are not multiplied together, but in the third equation e⁻ˣ which is not the first degree, the case is different as

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step-by-step explanation: