Series and Laplace

Question 1

When is it appropiate to use partial fractions?

Answer 1

When the bottom half of the fraction is differentiated and cannot be multiplied by any number (or manipulated in any way) to match the function on the top of the fraction.

Question 2

How does first shift theorem work?

Answer 2

For the inverse laplace, take the value of a given in the formula as the number affecting the s value on top of the fraction and change s-a to s. For laplace, seperate the exponential value and change s to s-a.

Question 3

How to solve differential equations using laplace (including simultaneous)?

Answer 3

Solve each variable to obtain F(s) values (F(s) typically shown as X or Y). The right side of the equation you should solve with laplace to get an s value. Rearrange to get F(s) on one side and inverse laplace the s values on the other side. For simultaneous equations, obtain X and Y values for both equations and use elimination method afterwards.

Question 4

How do you represent small and large values for x with taylor series?

Answer 4

Rearrange the function to a Maclaurin standard function. In this case you can divide out the number or the function from the function. E.g: (x+3)^-2 can be written as either 3^-2(1+x/3)^-2 for small x values or x^-2(1+3/x)^-2 for large values.

Question 5

How do you use taylor series for a value about a number (e.g: e^x about x=2)

Answer 5

Use basic principles to sub a = 2 (using the example) and always solve the derivative before using the a value. For this example f^’(a) = e² as f^’(x)=e^x.

Question 6

Explain the l’hospital rule

Answer 6

The fraction given has seperate functions to differentiate (no quotient rule) and you sub the value that x (for f(x) and g(x) as an example) tends to as the limit. Check the value for f^’/g^’ and differentiate further if no value is obtained. You keep differentiating and subbing the value in until you obtain a value or if it becomes impossible (e.g: differentiating past a constant like 5).

Question 7

For Fourier’s series how can you tell if a function is odd or even?

Answer 7

The graph of an even function is always symmetrical about the y-axis (i.e. it is a mirror image). The graph of an odd function is always symmetrical about the origin.

Question 8

Method of separation of variables for solving partial differential equations

Answer 8

First step is to try to make the function equal to the laplacian counterparts. For example: u(x,t) = X(x)T(t)

Second step is to differentiate still using those symbols. For example partial d²u/dt² would become X(x)T^’’(t).

Third step is to seperate variables if you need to (the constant should not go onto the ‘y’ term which in the example used would be x). After this each seperated term is then equal to a negative constant λ. Rearrange to obtain ODE for X(x) and T(t) as the example (both equal to zero).

Fourth step is to solve for a general solution. In the example we would try X(x) = e^mx and obtain a cos/sin solution. Same with T(t). Then replace the λ by making cos(λl) = 0 or sin(λl) = 0 depending on which constant cancels out to zero. λℓ = πn for sin(λl)=0 and λℓ = π/2 + nπ for cos(λl)=0. State that it equals to λ_n.

Fifth step is to use the boundary conditions to solve A,B,C,D (constants of the cos and sin terms). When you put all the terms together for u(x,t) = both general solutions multiplied together, new constants called α_n and β_n are formed which can be solved by a set of initial conditions. Make it into a series by putting a sum of symbol on the outside starting with n=1 to infinity. If the boundary codnition has a subscript of a function (like u_t(x,0)) then it requires differentiation of u(x,t).

Boundary conditions are what you use for in the ODE stages and initial conditions are used to determine the special constants (alpha and beta ones). Intial conditions for the example would start as u(x,0) whereas boundary conditions would start off as u(0,t).