Understanding Second Order Koshi Oiler Differential Equations

This video tutorial explains how to solve an initial value problem involving a second order Koshi Oiler differential equation. It covers the use of auxiliary equations to determine the general solution, including cases with distinct real roots, equal real roots, and complex roots. The tutorial demonstrates setting up the auxiliary equation, solving it, and finding the particular solution using given initial conditions. The process involves solving a system of equations to determine constants, leading to the final particular solution.

What is the primary characteristic of a second order Koshi Oiler differential equation?

• The degree of the coefficient is greater than the order of the derivative.

• The degree of the coefficient is unrelated to the order of the derivative.

• The degree of the coefficient equals the order of the derivative.

• The degree of the coefficient is less than the order of the derivative.

What is the purpose of the auxiliary equation in solving differential equations?

• To find the particular solution directly.

• To eliminate complex roots.

• To determine the initial conditions.

• To help find the general solution.

Which form of the general solution is used when the auxiliary equation has two distinct real roots?

• C1 * cos(mx) + C2 * sin(mx)

• C1 * e^(m1*x) + C2 * e^(m2*x)

• C1 * x^m + C2 * ln(x) * x^m

• C1 * x^m1 + C2 * x^m2

What are the values of a, b, and c in the auxiliary equation setup?

• a = 1, b = -7, c = 15

• a = 1, b = 7, c = -15

• a = -1, b = 7, c = 15

• a = 1, b = -7, c = -15

What are the roots of the auxiliary equation in this example?

• m1 = -5, m2 = -3

• m1 = 5, m2 = 3

• m1 = 3, m2 = 5

• m1 = -3, m2 = -5

What initial conditions are used to find the particular solution?

• y(0) = 31, y'(0) = 7

• y(1) = 31, y'(1) = 7

• y(1) = 7, y'(1) = 31

• y(0) = 7, y'(0) = 31

What is the derivative of the general solution y(x) = C1 * x^5 + C2 * x^3?

• 5 * C1 * x^5 + 3 * C2 * x^3

• C1 * x^5 + C2 * x^3

• 5 * C1 * x^4 + 3 * C2 * x^2

• C1 * x^4 + C2 * x^2

How is the value of C2 determined in the system of equations?

• By adding the equations.

• By subtracting the equations.

• By multiplying the equations.

• By substituting C1 = 7 - C2.

What is the final value of C1 in the particular solution?

• C1 = 2

• C1 = 5

• C1 = 7

• C1 = 3

What is the particular solution obtained in this example?

• y(x) = 2 * x^5 + 5 * x^3

• y(x) = 2 * x^3 + 5 * x^5

• y(x) = 5 * x^5 + 2 * x^3

• y(x) = 5 * x^3 + 2 * x^5