Understanding Koshi Oiler Differential Equations

This video tutorial explains how to solve a second order homogeneous Koshi Oiler differential equation. It begins by identifying the equation's components and coefficients, then uses an auxiliary equation to solve it. The quadratic formula is applied to find complex solutions, which are used to formulate the general solution. The tutorial emphasizes understanding the relationship between the equation's terms and the form of the solution.

What type of differential equation is discussed in the video?

• First-order homogeneous

• First-order non-homogeneous

• Third-order homogeneous

• Second-order homogeneous

What characteristic defines a Koshi Oiler differential equation?

• The degree of the coefficient is equal to the order of the derivative

• 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 is less than the order of the derivative

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

• a = 1, b = 7, c = -41

• a = 1, b = -7, c = 41

• a = 2, b = 8, c = -42

• a = 2, b = -8, c = 42

Why is the quadratic formula used in solving the auxiliary equation?

• The equation is not factorable

• The equation is linear

• The equation has no real roots

• The equation is factorable

What is the nature of the roots obtained from the auxiliary equation?

• No roots

• Two equal real roots

• Two distinct real roots

• Complex roots

What is the value of alpha in the general solution?

• 5

• 3

• 4

• 6

What is the value of beta in the general solution?

• 6

• 5

• 4

• 3

Which function is used in the general solution to represent the complex part?

• Tangent

• Exponential

• Sine

• Cosine

What mathematical operation is applied to x in the general solution?

• Exponential

• Square root

• Factorial

• Natural logarithm

What is the form of the general solution when complex roots are present?

• y(x) = C1 x^alpha + C2 x^beta

• y(x) = x^alpha (C1 cos(beta ln(x)) + C2 sin(beta ln(x)))

• y(x) = C1 e^(alpha x) + C2 e^(beta x)

• y(x) = C1 ln(x) + C2 ln(x)