2.3 Particular soln to some higher order eqn

Question 1

Linear ODE can be solved with method of undetermined coefficients if

Answer 1

1. q(t)’s derivatives spans finitely many functions
2. The linear ODE is a constant coefficient equation

Question 2

How to solve linear constant coefficient eqn with method of undetermined coefficients for GENERAL SOLUTION?

Answer 2

Particular Solution: Guess each possible derivative of q(t) multiplied by a constant (A,B,C, etc) . Plug in the guess and its derivatives back into the equation and then match the left hand side to the right to figure out what your guess’s constants are.
General Soln: Guess e^rt, see other card for multiplicities and imaginaries.

Question 3

How to solve cauchy-euler eqns for particular soln?

Answer 3

Complex x(t) = x1v1 + x2v2
OR -x1v1 + x2v2 if your v1 doesn’t include the negative sign already

Question 4

What if your particular solution is aready a homogeneous soln to a linear ODE?

Answer 4

Well, from variation of parameters you should multiply it by t.

Question 5

What is the equation for the wronskian?

Answer 5

x1 * x2’ - x1’ * x2 AKA det of the matrix with row1 x1,x2 and row2 is their derivatives.

Question 6

Assuming that the x’’ coefficient is never zero in an interval, then the wronskian according to a pair of homog. soln has the property:

Answer 6

either identically 0 or never 0.

Question 7

Cor: Two homog colutions are linearly independent IFF

Answer 7

IFF their wronskian is never zero on the interval of t.

Question 8

Solving for particular solutions of linear ODEs using variation of parameters?

Answer 8

Use the equation x1v1 + x2v2, where
v1 = - swint( (x2 * q) / (a2 * W) )
v2 = int( (x1 * q) / (a2 * W) )
where x1, x2 are the homog. solns, a2 is the leading function of t mult by x’’. W is wronskian.