Understanding First Order Linear Differential Equations

What is the defining characteristic of a first order linear differential equation?

• It can be written in the form y' + p(x)y = q(x).

• It includes a second derivative.

• It is always non-linear.

• It has a constant solution.

In the first example, what is the integrating factor used?

• e^(2x)

• e^(-2x)

• e^(x)

• e^(-x)

Why do we multiply the differential equation by the integrating factor?

• To transform the left side into a product rule form.

• To eliminate the derivative.

• To make the equation non-linear.

• To simplify the equation to a constant.

What substitution is used in the integration process of the first example?

• u = 2x

• u = -2x

• u = x

• u = -x

What is the general solution form for the first example?

• y = -3 + e^(2x)

• y = -3 + c * e^(2x)

• y = 3 + c * e^(2x)

• y = 3 + e^(2x)

In the second example, what transformation is applied to the equation initially?

• Dividing by x

• Adding a constant

• Multiplying by x

• Subtracting a constant

What is the integrating factor for the second example?

• 1/x

• e^(-x)

• x

• e^(x)

How is the left side of the equation transformed using the integrating factor in the second example?

• Into a difference

• Into a sum

• Into a product rule form

• Into a constant

What is the general solution form for the second example?

• y = 2x^2 + c/x

• y = 2x^2 - c/x

• y = 2x^2 - c

• y = 2x^2 + c

Why can't a particular solution be found in these examples?

• The equations are not solvable.

• The integrating factor is incorrect.

• The equations are non-linear.

• No initial conditions are provided.