When to use certain methods to differentiate

[drsmoothe2004]

i have a final coming up on wednesday and my professor will post 8 different differential equations without telling us which method to use. i want to know (generally speaking of course) which method would work for certain types of second order differential equations. for instance, judicious guessing is used to solve inhomogeneous 2nd order linear differential equations and variation of parameters is used to solve non constant coefficient inhomogeneous 2nd order linear differential equations.

1. my question is when should i be using series solutions and laplace transform to solve differential equations?

2. when using laplace transform to solve equations, is it crucial to be given initial conditions as well?

3. when solving systems of differential equations using matrices and eigenvalues, how to i put a equation into the form of $$\dot{x}$$ = (matrix A) multiplied by vector x and when will i know when to use this method? i would have to be given 2 differential equations correct?

4. last question: can someone explain the idea of the heavyside function with a shift at f(t)(t-t₀). i just don't quite get this whole shifting at f(t). and how to convert a piecewise function into the heavyside function?

examples of each would be excellent, and i just wanted to say thank you for taking the time to answer these questions. i find it in that many texts focus on what methods to use, but not enough focus on when to apply them.

 

[HallsofIvy]

1. Laplace transforms can only be used to solve linear equations with constant coefficients. Series solutions work for linear equations with variable coefficients.

2. Not necessarily. If you are not given y(a) or y'(a), leave them as unknown constants. After you have used Laplace transform to find the solution use whatever other conditions you are given to to determine those constants.

3. Are you thinking specifically of 2 by 2 matrices? n first order linear differential equations can be written as an n by n matrix equation with each row of the matrix corresponding to one equation. Also a single nth order linear differential equation can be written as system of n equations by introducing new variables for the first, second, ..., n-1 derivatives.

4. I don't know what you mean by "shift at f(t)(t- t₀". H(t) is defined as the function that is 0 for t< 0 and 1 for $0\le t$. H(t-t<sup>0[/sub]) is 0 for t< t₀ and 1 for [itex]t₀\le t[/itex].

To write a piecewise defined function as a sum of Heaviside functions, work from the left to the right. For example, suppose f(x) is defined to be $x^2$ for $x\le 0$, 1 for $0< x\le 1$, 2x for 1< x.

Since we want $f(x)= x^2$ for $x\le 0$ we start from that, without any Heaviside function. For 0< x, we want f(x)= 1. We can get that by just 1H(x) but if we just wrote $f(x)= x^2+ H(x)$ we would still have that "$x^2$" so we have to subtract that off: $f(x)= x^2+ (1- x^2)H(0)$. Now, for 1< x we want f(x)= 2x so we have to add 2xH(x-1). But, we still have to subtract off the "1" we just added: $f(x)= x^2+ (1- x^2)H(x)+ (2x- 1)H(x-1)$. For each step, you have to subtract off the previous step.

Look what happens: if $x\le 0$ both H(x) and H(x-1) are 0 so $f(x)= x^2$. If $0< x\le 1$ H(x)= 1 but H(x-1) is still 0: $f(x)= x^2+ (1- x^2)= 1$. Finally, if [1< x both H(x) and H(x-1) are 1: $f(x)= x^2+ (1- x^2)+ (2x- 1)= 2x$.</sup>

 

[drsmoothe2004]

when would it be wise to use eigenvalues and eigenvectors to solve differential equations?

i have a feeling for my exam, we will not be given a problem in matrix form such as

$$\dot{x}$$ = [1 2]
...[3 2] (2x2 matrix) multiplied by the vector x

but rather we will be given an equation such as 2y''-5y'+5=0

how does one know when to use laplace transform to solve such equations instead of solving for eigenvalues and eigenvectors to finding the general solution to the problem?

 

[HallsofIvy]

It is never necessary to use the Laplace transform and I consider it a waste of time. You should be able to take a problem like 2y"- 5y'+ 5= 0, write down its characteristic equation by sight, solve thatm, and immediately write the general solution.

 

[drsmoothe2004]

so, just to clarify, there is no specific time to use eigenvalues or vectors to solve second order differential equations?