Differential Equations and Roots

This video tutorial covers the process of solving higher order linear homogeneous differential equations with constant coefficients. It includes examples of third and fourth order equations, demonstrating solutions with distinct real roots, repeated roots, and complex roots. The tutorial explains the use of characteristic equations, factoring, and synthetic division to find solutions, and provides general solutions for each case.

What is the first step in solving a higher order constant coefficient linear homogeneous differential equation?

• Find the particular solution.

• Assume the solution is of the form y = e^(rx).

• Directly integrate the equation.

• Use numerical methods to approximate the solution.

In the example of the third-order differential equation, what is the characteristic equation derived?

• R^3 - R^2 + 3R - 3 = 0

• R^3 - 3R^2 - R + 3 = 0

• R^3 + 3R^2 - R - 3 = 0

• R^3 - 3R^2 + R + 3 = 0

How many distinct real roots were found in the third-order differential equation example?

• One

• Four

• Two

• Three

What is the general solution form for a differential equation with three distinct real roots?

• y(x) = C1 e^(r1x) + C2 e^(r2x) + C3 sin(r3x)

• y(x) = C1 e^(r1x) + C2 x e^(r2x) + C3 x^2 e^(r3x)

• y(x) = C1 e^(r1x) + C2 e^(r2x) + C3 e^(r3x)

• y(x) = C1 e^(r1x) + C2 e^(r2x) + C3 x e^(r3x)

In the fourth-order differential equation example, what is the multiplicity of the root found?

• One

• Four

• Three

• Two

What is the general solution form for a differential equation with a root of multiplicity three?

• y(x) = C1 e^(x) + C2 e^(2x) + C3 e^(3x)

• y(x) = C1 + C2 e^(x) + C3 x e^(x) + C4 x^2 e^(x)

• y(x) = C1 + C2 e^(x) + C3 e^(2x)

• y(x) = C1 e^(x) + C2 x e^(x) + C3 x^2 e^(x)

What is the characteristic equation for the third-order differential equation with complex roots?

• R^3 - 2R^2 + 9R - 18 = 0

• R^3 + 2R^2 - 9R + 18 = 0

• R^3 - 2R^2 - 9R + 18 = 0

• R^3 + 2R^2 + 9R - 18 = 0

What are the complex roots found in the third-order differential equation example?

• R = ±5i

• R = ±4i

• R = ±2i

• R = ±3i

What is the general solution form for a differential equation with complex roots?

• y(x) = C1 e^(αx) cos(βx) + C2 e^(αx) sin(βx)

• y(x) = C1 e^(αx) + C2 e^(βx)

• y(x) = C1 cos(αx) + C2 sin(βx)

• y(x) = C1 e^(αx) + C2 x e^(βx)

In the example with complex roots, what is the value of alpha in the general solution?

• 0

• 1

• 2

• 3