Introduction & Terminology Flashcards
(21 cards)
What is a differential equation?
It’s an equation (identity) where derivatives are involved
What type of solutions can we have in differential equations?
General ones (involving a general constant C) or particular ones which appear when initial value problem are involved.
What is the order of a differential equation?
It is the highest derivative that appears in the equation
What is the separation of variables method about?
It’s a method for solving differential equations which consists in getting together functions of certain variables with their respective differentials
F(y)dy = g(x)dx
What’s a trivial solution to a differential equation?
It’s when y = 0 represents a solution to the equation.
What is the geometric meaning of a differential equation?
Slope fields, these are just slopes all around the XY plane
How are called the the curves that follow the paths of a slope field?
Isoclines or integral curves
When can we assure that a differential equation has a unique solution to the initial value problem at (xo, yo)?
When F(x, y) and the partial derivative of F(x, y) with respect to y are continuous near (xo, yo).
What’s a linear differential equation?
A differential equation where none of the derivatives have a power different than one. Example:
y’ + P(x)y = Q(x)
How do you solve a linear differential equation?
By calculating the integrating factor:
exp(Integral(P(x)dx))
What is the Bernoulli’s equation?
It’s a linear differential equation where the expression Q(x) is being multiplied by y to the n.
To solve it, we divide by the aforementioned term and perform a change of variable
When is a differential function homogeneous?
Whenever:
f(tx, ty) = f(x, y)
How do you solve a homogeneous differential equation?
By making a substitution:
y = vx (which implies)
dy = vdx + xdv
What should I do for solving this D.E.
f(x, y) = f(ax +by + c)
To perform the substitution that will transform the D.E. Into separable
z = ax + by + c
dz = adx + bdy
How to reduce to homogeneous the following:
(ax + by + c)dy + (ex + fy + g)dx
If the lines intersect at xo and yo, we do:
X = x + xo & Y = y + yo
If they don’t intersect, we do a substitution z = ax + by (it works for both of them)
When is a differential equation said to be exact? Consider it in this form?
M(x,y)dx + N(x,y)dy = 0
When the partial derivative of M with respect to y is the same as the partial derivative of N with respect to x.
What’s an autonomous differential equation?
It’s a differential equation that depends only on one variable, more specifically, on the dependent variable.
For instance:
dy/dt = f(y)
How are called the values that make zero an autonomous differential equation?
They’re called equilibrium solutions of the differential equation.
What are asymptotic and unstable equilibrium solutions of autonomous differential equations?
Asymptotic are those that that start at a point and tend towards the equilibrium solutions.
An unstable one is one that starts close to the equilibrium solutions and tends towards infinity.
What is the general form a of logistic equation?
dy/dt = ay(1 -y/k)
y : Total population
k : Carrying capacity
Equilibrium: y = 0 or y = k
What should we do for reducing to an exact differential equation?
(aM/ay - aN/ax)/N should depend on x
(aN/ax - aM/ay)/M should depend on y
