Understanding Linear First Order Equations and Integrating Factors

What is the normal form of a linear first order equation?

• y' + p(x)y = q(x)

• y'' + p(x)y = q(x)

• y' = p(x) + q(x)y

• y = p(x)y' + q(x)

In a linear first order equation, what can p(x) and q(x) be?

• Functions of both x and y

• Only constants

• Functions of y

• Functions of x or constants

What is the purpose of using an integrating factor in solving linear first order equations?

• To make the equation non-linear

• To find the derivative of y

• To eliminate the y term

• To convert the equation into a product rule

How is the integrating factor for a linear first order equation determined?

• e to the power of y

• e to the integral of p dx

• e to the integral of q dx

• ln of the integral of p dx

In the example y' + y = e^x, what is the integrating factor?

• x

• 1

• e^(2x)

• e^x

What is the solution to the equation y' + y = e^x after applying the integrating factor?

• y = e^x + C

• y = 0.5e^x + Ce^(-x)

• y = e^(2x) + C

• y = x + Ce^x

In the equation x^2y' + 5xy = x, what is the first step to solve it using the integrating factor method?

• Divide by x^2

• Multiply by x^2

• Add 5xy to both sides

• Subtract x from both sides

What is the integrating factor for the equation x^2y' + 5xy = x after converting it to normal form?

• x^5

• e^(5x)

• x^2

• e^(x^2)

What is the general form of the integrating factor for linear first order equations?

• e to the integral of p dx

• e to the integral of q dx

• ln of the integral of q dx

• ln of the integral of p dx

What is the key step to ensure before finding the integrating factor?

• Equation has no constant terms

• Equation is in normal form

• Equation is quadratic

• Equation is non-linear