Chapter 1: First Order ODEs

Question	Answer
Cauchy Problem	ODE given in explicit form with initial conditions
Homogeneous function of degree k	A function such that h(vx, vy) = v^k h(x,y) Always reduce to separable
Bernoulli Equation	y' + p(x)y = g(x)y^a
Solution of Bernoulli Equation	Let u = y^(1-a) Differentiate and substitute into equation
Exact ODE	An ODE in the form M(x,y)dx + N(x,y)dy = 0 if there exists a function u such that M(x,y)dx + N(x,y)dy is the total differential of u
Criterion for Exactness	dM/dy = dN/dx Mixed partial of u must be equal to satisfy continuity
Solution of Exact ODEs	u(x,y) = \int{M}dx + k(y) where k(y) serves as the constant of integration then du/dy = N and solve for k(y)
Integrating factor	For a non-exact equation Pdx + Qdy = 0 A function F such that FPdx + FQdy = 0 is exact
Finding Integrating factors	X(w) = (P_y - Q_x)/(Qw_x - Pw_y) Find an w such that X(w) is a function of only x or y F = exp(\int X(w) dw)
Equilibrium state	The point at which the main derivative is 0
Relaxation equation	A first order ODE in which the terms "counteract" each other such that at some point equilibrium is reached

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	Created by Sam Wilson over 4 years ago