General Solutions of Differential Equations

The video tutorial explains how to find the general solution to a fourth order constant coefficient linear homogeneous differential equation. It begins by introducing the concept and then moves on to finding the characteristic equation and its roots. The tutorial covers both distinct and repeated roots, using them to derive the general solution. The explanation is based on knowledge of second order equations and includes step-by-step guidance on factoring and solving the characteristic equation.

What type of differential equation is being solved in the video?

• Third order variable coefficient

• Fourth order constant coefficient linear homogeneous

• Second order non-linear

• First order linear

What is the characteristic equation derived from the differential equation?

• R^2 - 5R + 6 = 0

• R^3 - 5R^2 + 6R = 0

• R^4 - 5R^3 + 6R^2 = 0

• R^4 + 5R^3 - 6R^2 = 0

How many roots does the characteristic equation have?

• Four

• Three

• Two

• One

What is the multiplicity of the root R = 0?

• 4

• 2

• 1

• 3

Which of the following is a distinct root of the characteristic equation?

• R = 0

• R = 1

• R = 2

• R = 4

What is the general solution form for two distinct real roots in a second order equation?

• y(x) = c1 * e^(R * x) + c2 * x * e^(R * x)

• y(x) = c1 * e^(R1 * x) + c2 * x^2 * e^(R2 * x)

• y(x) = c1 * e^(R1 * x) + c2 * e^(R2 * x)

• y(x) = c1 * x * e^(R1 * x) + c2 * e^(R2 * x)

What additional factor is included in the solution for repeated roots?

• A quadratic term

• A constant

• An exponential term

• A linear term

Which root is used with multiplicity 2 in the final solution?

• R = 0

• R = 3

• R = 1

• R = 2

What is the simplified form of e^0 in the general solution?

• 0

• 1

• e

• x

What is the final form of the general solution?

• y(x) = c1 * e^(2x) + c2 * e^(3x) + c3 * x + c4 * x^2

• y(x) = c1 * e^(2x) + c2 * e^(3x) + c3 + c4 * x

• y(x) = c1 * e^(2x) + c2 * e^(3x) + c3 * x^2 + c4

• y(x) = c1 * e^(2x) + c2 * e^(3x) + c3 * x + c4