Differential equations

Question 1

First thing to look at when solving odes

Answer 1

is it separable: can move variables completely to either side of the =’s

Question 2

General solution of an ode

Answer 2

involves constants that have not been determined

Question 3

Particular solution of an ode

Answer 3

With initial values we can determine the values of the constants from the general solution.

Question 4

Solving first order

Answer 4

Make sure dy/dx is not multiplied by anything then use p(x) in the integrating factor.
u(x) =e^∫p(x)dx
Then y = (∫q(x)u(x)dx)/u(x)

Question 4

1st order form:

Answer 4

dy/dx +p(x)y = q(x)

Question 5

Form for 2nd order

Answer 5

y’’ +p(x)y’ + q(x)y = f(x)

Question 6

How to tell if 2nd order is homogenous or not?

Answer 6

if f(x) = 0 then homogenous

Question 7

What to do if differential equation is separable?

Answer 7

Separate the variables and solve accordingly.

Question 8

Solving homogenous 2nd order

Answer 8

Sub in y=e^kx, so equation will go to a quadratic, solve the quadratic and it will either be 2 real roots, one real repeated root or no real roots (use i), each of which will have a solution we can use these to determine general solution.

Question 9

Solution for 2 real distinct roots

Answer 9

y = C1e^k1x+C2e^k2x
Where k1 and k2 are solutions to the quadratic and C1 and C2 are constants

Question 10

Solution for one repeated root

Answer 10

y= C1e^kx+C2xe^kx, where C1 and C2 are constants and k is our repeated root

Question 11

Solution for no real roots with imaginary numbers

Answer 11

y = e^ax(C1cos(bx)+C2sin(bx))
For solution:
a ± bi

Question 12

Solving non-homogenous 2nd order differential equations initial steps

Answer 12

Initially evaluate as though homogenous until we have calculated solution for that then solve with f(x)

Question 13

Guess equation for f(x) = ne^rx

Answer 13

f(x) = Ae^rx

Question 14

Guess for f(x) = polynomial degree n

Answer 14

f(x) = ax^n + bx^n-1….. until exponent = 0

Question 15

guess for f(x) triganometric polynomial
e.g. nSin(rx)

Answer 15

f(x) = Acos(rx) +Bsin(rx)

Question 16

What happens if in our homogenous solution we have the same exponent to e as our guess for f(x)

Answer 16

We make our guess Axe^rx

Question 17

Steps for solving latter half of non homogenous equation

Answer 17

Once we have our yh (homogenous), we need to choose an appropriate guess for what our f(x) should look like so we use very general ones depending on the type with constants.
We then work out the derivates, substitute those in and solve for the constants, once calculated we put those in our initial guess. This will then be added to the homogenous part to give the overall solution