This video tutorial covers solving linear second order homogeneous differential equations with constant coefficients, focusing on cases where the characteristic equation has two distinct real roots. It includes a review of the form of these equations, the characteristic equation, and the general solution. Three examples are provided: one with distinct real roots, another with a zero coefficient, and a third using the difference of squares. The video concludes with a note on other types of roots to be covered in future lessons.

What type of roots does the characteristic equation have in the examples discussed in the video?

• Two distinct real roots

• Imaginary roots

• Repeated real roots

• Complex roots

Tags

CCSS.8.EE.C.8B

What is the form of the characteristic equation for a linear second order homogeneous differential equation with constant coefficients?

• a r^2 + b r + c = 0

• a r^2 + b = 0

• a r + c = 0

• b r^2 + c = 0

In the first example, what are the values of the coefficients a, b, and c?

• a = 1, b = -1, c = 0

• a = 0, b = -1, c = -6

• a = 1, b = 0, c = -6

• a = 1, b = -1, c = -6

What is the general solution form for the first example's differential equation?

• y(x) = c1 e^(2x) + c2 e^(-3x)

• y(x) = c1 e^(4x) + c2 e^(-4x)

• y(x) = c1 e^(3x) + c2 e^(-2x)

• y(x) = c1 e^(x) + c2 e^(-x)

Tags

CCSS.8.EE.C.7B

In the second example, what is the value of c?

• c = -6

• c = 1

• c = 0

• c = -4

Tags

CCSS.8.EE.C.7B

What is the general solution for the second example's differential equation?

• y(x) = c1 + c2 e^(4x)

• y(x) = c1 e^(2x) + c2 e^(-2x)

• y(x) = c1 e^(4x) + c2

• y(x) = c1 e^(x) + c2 e^(-x)

Tags

CCSS.8.EE.C.7B

In the third example, what is the value of b?

• b = 9

• b = 0

• b = 1

• b = -4

Tags

CCSS.8.EE.C.7B

What is the general solution for the third example's differential equation?

• y(x) = c1 e^(2/3 x) + c2 e^(-2/3 x)

• y(x) = c1 e^(4x) + c2 e^(-4x)

• y(x) = c1 e^(3/2 x) + c2 e^(-3/2 x)

• y(x) = c1 e^(x) + c2 e^(-x)

What is the form of the general solution when the characteristic equation has two distinct real roots?

• y(x) = c1 e^(r1 x) + c2 e^(r2 x)

• y(x) = c1 e^(r1 x) + c2

• y(x) = c1 + c2 e^(r2 x)

• y(x) = c1 e^(x) + c2 e^(-x)

What will be covered in future examples according to the video?

• Examples with polynomial roots

• Examples with no roots

• Examples with imaginary roots

• Examples with complex roots