Differential equations Flashcards
First thing to look at when solving odes
is it separable: can move variables completely to either side of the =’s
General solution of an ode
involves constants that have not been determined
Particular solution of an ode
With initial values we can determine the values of the constants from the general solution.
Solving first order
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)
1st order form:
dy/dx +p(x)y = q(x)
Form for 2nd order
y’’ +p(x)y’ + q(x)y = f(x)
How to tell if 2nd order is homogenous or not?
if f(x) = 0 then homogenous
What to do if differential equation is separable?
Separate the variables and solve accordingly.
Solving homogenous 2nd order
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.
Solution for 2 real distinct roots
y = C1e^k1x+C2e^k2x
Where k1 and k2 are solutions to the quadratic and C1 and C2 are constants
Solution for one repeated root
y= C1e^kx+C2xe^kx, where C1 and C2 are constants and k is our repeated root
Solution for no real roots with imaginary numbers
y = e^ax(C1cos(bx)+C2sin(bx))
For solution:
a ± bi
Solving non-homogenous 2nd order differential equations initial steps
Initially evaluate as though homogenous until we have calculated solution for that then solve with f(x)
Guess equation for f(x) = ne^rx
f(x) = Ae^rx
Guess for f(x) = polynomial degree n
f(x) = ax^n + bx^n-1….. until exponent = 0
guess for f(x) triganometric polynomial
e.g. nSin(rx)
f(x) = Acos(rx) +Bsin(rx)
What happens if in our homogenous solution we have the same exponent to e as our guess for f(x)
We make our guess Axe^rx
Steps for solving latter half of non homogenous equation
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
