Understanding Linear First Order Differential Equations

The video tutorial explains how to solve a linear first-order differential equation with an initial condition. It covers the steps to put the equation in the correct form, identify the functions P(T) and F(T), and determine the intervals where these functions are continuous. The tutorial concludes by finding the interval in which the solution to the initial value problem is certain to exist.

What is the initial condition given in the problem?

• y(2) = -1

• y(-2) = -1

• y(2) = 1

• y(-2) = 1

What must be true for P(T) and F(T) to ensure a unique solution exists?

• They must be equal.

• They must be continuous on an open interval.

• They must be continuous on a closed interval.

• They must be differentiable.

What is the first step in rewriting the differential equation?

• Divide by T-1

• Multiply by T-1

• Add T-1

• Subtract T-1

What is P(T) in the rewritten differential equation?

• 6t / (T-1)

• ln(T+3) / (T+1)

• ln(T+3) / (T-1)

• 6t / (T+3)

What condition must be met for the natural log input in P(T)?

• T must be less than -3

• T must be less than 3

• T must be greater than -3

• T must be equal to -3

What is the interval where P(T) is continuous?

• (-3, ∞)

• (-∞, 3)

• (-∞, 1) U (1, ∞)

• (-3, 1) U (1, ∞)

What is the only restriction on F(T)?

• T cannot be 3

• T cannot be 1

• T cannot be 0

• T cannot be -3

What is the interval of validity for the solution?

• (-3, ∞)

• (-∞, 3)

• (-∞, 1) U (1, ∞)

• (-3, 1) U (1, ∞)

Which interval contains the initial condition value of -2?

• (-∞, -3)

• (-3, 1)

• (1, ∞)

• (-∞, 1)

What is the final interval where the solution is certain to exist?

• T > -3 and T < 1

• T > 1

• T < -3

• T > 3 and T < 1