Differential Equations and Initial Conditions

The video tutorial explains how to solve a system of differential equations with given initial conditions. It begins by introducing the equations and differentiating them to form a second-order linear homogeneous differential equation. The characteristic equation is derived and solved to find the roots, which are used to construct the general solution for y(t). The video then calculates x(t) using y(t) and its derivative. Initial conditions are applied to determine the constants C1 and C2, leading to the final solution of the system. The tutorial concludes with a summary of the solution process.

What are the initial conditions given for the functions x(t) and y(t)?

• x(0) = 0, y(0) = 2

• x(0) = 3, y(0) = 1

• x(0) = 1, y(0) = 3

• x(0) = 2, y(0) = 4

What is the form of the second-order linear homogeneous differential equation derived?

• y'' = 3y' + 2y

• y'' = y' + 2y

• y'' = 2y' + y

• y'' = y' + 3y

What is the characteristic equation obtained from the differential equation?

• r^2 + r - 2 = 0

• r^2 - 2r + 1 = 0

• r^2 + 2r - 1 = 0

• r^2 - r - 2 = 0

What is the general solution for y(t) based on the roots of the characteristic equation?

• y(t) = C1 e^(t) + C2 e^(t)

• y(t) = C1 e^(-t) + C2 e^(2t)

• y(t) = C1 e^(2t) + C2 e^(-t)

• y(t) = C1 e^t + C2 e^(-2t)

How is x(t) determined from y'(t)?

• x(t) = 0.5y'(t)

• x(t) = 2y'(t)

• x(t) = y'(t)

• x(t) = y'(t) - 1

What is the expression for y'(t) in terms of C1 and C2?

• y'(t) = 2C1 e^(t) - 2C2 e^(-t)

• y'(t) = C1 e^(2t) + C2 e^(-t)

• y'(t) = C1 e^(t) + C2 e^(-2t)

• y'(t) = 2C1 e^(2t) - C2 e^(-t)

What equation is used to find C1 using the initial condition x(0) = 1?

• C1 - C2 = 0

• C1 - 0.5C2 = 1

• C1 + C2 = 1

• C1 + 0.5C2 = 1

What equation is used to find C2 using the initial condition y(0) = 3?

• C1 + 2C2 = 3

• C1 - C2 = 3

• C1 + C2 = 3

• C1 - 0.5C2 = 3

What is the value of C1 after solving the system of equations?

• C1 = 1/3

• C1 = 4/3

• C1 = 5/3

• C1 = 2/3

What is the final expression for x(t) after substituting the constants?

• x(t) = 5/3 e^(t) - 1/3 e^(-2t)

• x(t) = 4/3 e^(2t) + 2/3 e^(-t)

• x(t) = 2/3 e^(2t) + 5/3 e^(-t)

• x(t) = 5/3 e^(2t) - 2/3 e^(-t)