Definitions Flashcards
Just to lead
Separable
can be formed into f(x)dx = f(y)dy
How to solve a separable ODE
integrate both sides. Note that a separable ODE is always easily solvable for y’, but if you cannot separate f(x,y) then it is something else
Homogenous
If f(xt, yt, y’) = tf(x,y,y’)
How to solve Homogenous ODEs
divide the whole equation by the variable that it is homogenous to, if it is homogenous to x and y, pick one. Afterward, you will always have to make the substitution z = f(x,y) usually z = x/y
Linear ODE
ODE linear wrt a variable AND its derivative
How to solve a Linear ODE
First isolate y’, then P/Q becomes your integrating factor, if necessary, otherwise solve directly
Bernoulli Equations
Super close to linear but… also n ≠ 0, 1
To solve a Bernoulli Equation
Depending which variable… then make the z substitution and start over
Riccati Equation
similar to linear, but there’s a squared dependent variable in a 4th term
How to solve Ricatti Equation
Look for a particular solution y1(x) = … then form your new y with that particular solution. When you substitute the new y, you can start over, but will usually have a Bernoulli
how do you solve an equation that is not a quasi polynomial?
You find the homogenous solution, afterwards, you form a particular solution with this as reference, and you do the wronskian ;D
how do you formulate yh for Euler form with distinct, real x
yh = c1x^r1 + c2x^r…
how do you formulate yh for Euler form with irrational, real x
yh = c2|x|^r1…
how do you formulate yh for Euler form with repeated, real x
yh = c1x^r + c2(ln|x|)|x|^r
simple method for variation of parameters (let C vary ♫ヽ(゜∇゜ヽ)♪♬(ノ゜∇゜)ノ♩♪)
generate your yp from yh and solve the system
