2.1&2.2 Soln to Some Homog Eqns

Question 1

Constant Coefficient Linear Equations has the form:

Answer 1

Each x’’’, x’’, x’, x, etc only has a constant in front of it

Question 2

How to solve constant coefficient linear equations

Answer 2

Guess solution x = e^rt, plug in the x and it derivatives, and get a polynomial must equal a 0. From there, determine the roots and those are your solutions if there are 2.

Question 3

What to do if linear coefficient solution roots have multiplicities or are imaginary?

Answer 3

1. If multiplicities, then use diff. of variables to get v(t) = (c.1 + t) AKA multiply it by t to get the second soln.
2. If imaginary, a +- bi gets the form c.1 * e^at * cos(bt) + c.2 * e^at * sin(bt).

Question 4

What is variation of parameters? When you have a homog soln and need another one.

Answer 4

Write your homog soln as x = v(t)* x.h1 AKA your guess * some function v. Then, plug in x and its derivatives and simplify until you find out that v’’ = 0 or v’’ = e^t and then differentiate twice to get v, and then multiply by your previous guess.

Question 5

Cauchy Euler Eqn has the form:

Answer 5

t^2 x’’ + tx’ + x = q(t) AKA each x derivative to degree n is multiplied by a t to the equivalent n power.

Question 6

How to solve cauchy euler eqns?

Answer 6

Guess x = t^r, sub in x and all its derivatives to get a polynomial equals 0. Then get the roots of r, and plug back into solution

Question 7

What to do if cauchy euler guess of x = t^r fails to provide as many roots as needed or they’re imaginary?

Answer 7

1. If they are multiplicities, multiply the guess by log(t) to get a new guess (AKA v = 1 + log(t))
2. If they are imaginary with a +- bi, then you can get the soln of the form: x = c1 * t^a * cos( blog(t) ) + c2 * t^a * sin( blog(t) ).

Question 8

What to do if linear coefficient nth order ODEs have imaginary roots? Such that it is r= a plusminus bi to the solns e^rt?

Answer 8

You do e^at * cos(bt) + e^at * sin(bt)

Question 9

Imaginary cauchy euler eqn solutions

Answer 9

R = a plusminus bi.
c1t^a * cos( b * log(t) ) + c2t^a * sin( b * log(t) )

Question 10

How do you solve first order inhomogeneous solutions (i.e. x’ + p(t) * x = q(t) )?

Answer 10

Integrating factor method!
1. u(t) = int(p(t))
2. Multiply equation by u(t)
3. Integrate to get solution, where LHS will collapse into a product rule.