MATH 320 Differential Equations Final Exam WCU

here exists a number A so that y=e^(5x+1)-e^(2x+2) solves the differential equation d^2y/dx^2+A(dy/dx)+6y=0

False, (5x * 2x) =10 has to equal 6

If f(t) and g(t) are two functions with respective Laplace transforms F(s) and G(s), then L{f(t)g(t)} = F(s)G(s).

False, L(f⋅g) = F(s)G(s)

There is no solution to the initial value problem dy/dx= y/x with y(1)=1 defined for all real numbers.

False, Undefined at x=0

Does there exist a continuous function g(y) so that the autonomous differential equation (dy/dx)=g(y) has multiple (at least two) equilibria, and all the equilibria are asymptotically stable

False, all the equilibria can't be stable

There is a unique solution tot he differential equation d^2y/dx^2=-y such that y(0)=0

False, because y must be greater than 0 meaning we can't find a solution.

(a) There exists a third-order homogeneous differential equation with constant coefficients which has y = cos(2x) + sin(x) as a solution.

False, the Coefficient the insides of the cos and sin function or it equal and there is only two parts to the solution

Since the inverse Laplace transform of 1/s^2 is t, the inverse Laplace transform of 1/(s-3)^2 must be t - 3.

False, the inverse Laplace transform is te^(3t) not t-3

If m and b are constant, then the differential equation (dy/dx)=my+b is solved by y=Ce^(bx)+m where C is a constant.

False, y=Ce^(mx)-b/m

There exists a number A so that y=e^(3x+1)-e^(2x+2) solves the differential equation d^2y/dx^2+A(dy/dx)+6y=0

True the 3x * 2x = the 6y component

The differential equation dy= 2x+1 dx 3y^2+y-2 Is implicitly soved by y^3+(y^2)/2-2y = x^2+x+C

True, 3y^2+y-2(dy)=2x+1 (dx)

The differential Equation cos(x)/x^2+(e^2)y=sec^3(x)

True, No y(dy)/(dx)

There exists a fourth-order homogeneous differential equation with constant coefficients which has y = cos(2x) + sin(x) as a solution. True, it (2 Cos (2x) - Cos(2x) + (25in (2x) - Sin (2x)

True, if (2 Cos (2x) - Cos(2x) + (25in (2x) - Sin (2x)

For all constants , and all functions f(t) and g(t) whose Laplace transform exists, we have L{f(t) + λg(t)} = L{f(t)} + λL{g(t)}

True, using Laplace transforms we can separate them this way

If t is measured in years and a quantity y changes in such a way that it halves every r years, then y satisfies the differential equation (dy)/(dt) =(-ln(2))/r)y

True, y=C(2^(ln(2)))

If a ±ib are complex conjugate roots of the characteristic polynomial of a homo-geneous differential equation with constant coefficients, which of the following are real valued solutions to the differential equation? a) e^(a+ib)x and e^(a−ib)x (b) e^ax cos(bx) and e^ax sin(bx) (c) cos((a + ib)x) and sin((a + ib)x) (d) e^a cos(ax) and e^b sin(bx)

b

Which of the following are equivalent to saying yı(x), y2(2), Yn (x) form a fundamental set of solutions to an n-th order homogeneous linear differential equation? (a) The Wronskian of the solutions is non-zero. (b) The solutions are linearly independent. (c) Both (a) and (b). (d) None of the above.

b

Which of the following techniques for solving a differential equation applies to day (d^2y/dx^2)+ x^2(dy/dx)+xy=0 (a) Integrating factor. (b) Method of undetermined coefficients. (c) Variation of parameters. (d) Power series.

b

How many linearly independent solutions are there to an n-th order homogeneous linear differential equation? (a) 1 (b) n −1 (c) n (d) It depends on the coefficients appearing in the differential equation.

c

Which of the following techniques for solving a differential equation applies to solving (d^2y/dx^2)+ 2(dy/dx) + 3y = ln(x2 + 1)? (a) Integrating factor. (b) Method of undetermined coefficients. (c) Variation of parameters. (d) Power series.

c

Which of the following expressions equals the Wronskian of two solutions y1 andy2 to a second-order linear homogeneous differential equation? (a) y1y′1 −y2y′2 (b) y1y′2 + y′1y2 (c) y1y′1 + y2y′2 (d) y1y′2 −y′1y2

d