Variation of Parameters Quiz

What does it mean for a differential equation to be non-homogeneous?

• The equation is not linear.

• The right-hand side is not zero.

• The right-hand side is zero.

• The coefficients are not constant.

In the variation of parameters method, what is the role of the Wronskian?

• It determines the order of the differential equation.

• It is used to solve the characteristic equation.

• It is used to find the complementary function.

• It helps in checking the linear independence of solutions.

What is the form of the particular solution in the variation of parameters method?

• c1y1 + c2y2

• u1y1 + u2y2

• y1 + y2

• c1 + c2

How do you find u1 and u2 in the variation of parameters method?

• By integrating their derivatives.

• By using the characteristic equation.

• By differentiating the complementary function.

• By solving a system of linear equations.

In the example problem, what is the characteristic polynomial of the associated homogeneous equation?

• M^2 + 1 = 0

• M^2 - 2 = 0

• M^2 - 1 = 0

• M^2 + 2 = 0

What are the values of M obtained from the characteristic equation in the example problem?

• 0 and 1

• i and -i

• 1 and -1

• 2 and -2

What is the Wronskian of the functions y1 = cos(x) and y2 = sin(x)?

• cos(x)sin(x)

• 0

• 1

• cos^2(x) + sin^2(x)

What is the particular solution Yp in the example problem?

• cos(x)ln(x) + sin(x)

• cos(x)ln(cos(x)) + xsin(x)

• cos(x) + sin(x)

• ln(cos(x)) + x

What is the general solution of the differential equation in the example problem?

• C1cos(x) + C2sin(x)

• C1cos(x) + C2sin(x) + cos(x)ln(cos(x)) + xsin(x)

• C1 + C2 + cos(x)ln(cos(x)) + xsin(x)

• C1cos(x)ln(cos(x)) + C2xsin(x)

In higher-order equations using variation of parameters, what is unique about the columns replaced in the Wronskian?

• They are identical to the original columns.

• They contain only zeros.

• They contain only ones.

• They have zeros except for the last entry.