Basic techniques for solving Ordinary Differential Equations
Is differential equation Linear?
With some manipulation, you can use Integrating Factor
Is it a Bernoulli equation?
No
Yes
Is it seperable?
Real, repeated roots
Is forcing function a "black sheep"?
Yes
BEWARE- SERPENTS HAVE BEEN SPOTTED IN THIS AREA!
No? You missed something, turn around.
Can you put it in standard form?
NO
Now, use Undetermined Coefficients to solve for particular solution
Take the Laplace transformation, solve for L(y), and take inverse Laplace, NOTE- You need initial conditions to solve completely.
Form the characteristic equation with powers of r that match orders of y as well as matching all constant coefficients. Then proceed to factoring.
Yes
Find p(t), muliply both sides by u(t), carry out integration, then solve for y.
Then you must use Variation of parameters
Integrating Factor
Yes
Yes
No
Use the form y=e^at(C1costbt+C2sinbt)
Imaginary roots
Does it include a step function?
Does it include a step function?
First find the homogeneous solution using the method below, then come back here.
yes
No
Is it homogeneous?
No, above 1st order
First find the homogeneous solution
Yes
Yes
Than you must take the Laplace transformation, solve for L(y), then take inverse Laplace. NOTE- you will not be able to entirely solve without initial conditions.
Use the form Y=C1e^rt+C2te^rt
Real independent roots
You could try converting in to a system of first order DE's, put in to a martrix, and solve using Eigenvectors
Is it a 1st order?
Use the form Y=C1e^r1t+C2e^r2t
Yes, seperate and solve directly
You will have to use Euler's Method to approximate a solution
