Week 10 ODE 1 Flashcards
what is the general solution of an ODE
set of all functions that satisfy it
what form do first order ODEs have
dy/dx = f(x, y)
what is the general equation for a separable ODE dy/dx = f(x)g(y)
∫ 1 / g(y) dy = ∫ f(x) dx
what is the general form of a first order linear ODE
dy/dx + P(x)y = Q(x)
what is the integrating factor equation
I = e^∫P(x)dx
what is the general form of an integrated first order linear ODE
y(x) = 1/I ∫QI dx
what can often be used to solve ODEs
a substitution
give one of the general forms that an ODE can written in
dy/dx = F(y/x) dy/dx = f(x,y) / g(x,y) g and f must be of the same degree
what is the substitution we use to make a homogenous ODE separable
y = uv
what is the general form of a Bernoulli equation
dy/dx + P(x)y = Q(x)y^n
what is the substitution used for a Bernoulli equation
u = y^-(n-1)
what substitution can be used to reduce a 2nd order ODE to a 1st order ODE
p = dy/dx
what is the general coefficient form of a second order ODE that can’t be reduced
a d^2y/dx^2 + b dy/dx + cy = f(x)
what is the 4 step procedure for solving a 2nd order ODE
1 consider when f(x) = 0
2 find the auxiliary equation
3 find the roots of the auxiliary equation
4 find the appropriate complementary function
what is the general form of the auxiliary equation
am^2 + bm + c = 0
