Higher Order Differential Equations Flashcards
(18 cards)
What is the form of a 2nd order linear differential equation?
y’’ + p(x)y’ + q(x)y = f(x)
Existence and uniqueness theorem for 2nd linear ODE?
We need p(x), q(x) and f(x) to e continuous in an interval I that contains the points related to the initial condition:
y(a) = b0 & y’(a) = b1
What is the superposition theorem?
If y1 and y2 are both solutions to the 2nd linear ODE, then the linear combination of them is also a solution.
When do we say that two solutions are linearly independent?
When there’s no way to write one of them as a scalar multiple of the other.
What happens if there are two solutions linearly independent of a 2nd order linear ode?
Then, the linear combination of them represents the general solution.
What is a constant coefficient linear homogeneous differential equation?
An equation of the following form:
ay’’ + by’ + cy = 0
How do we solve a constant coefficient linear homogeneous differential equation?
By guessing a solution of the form:
y = exp(rt)
When solving a second order constant coefficient linear homogeneous ode, what are my possibilities?
We can have two solutions that could be:
1) exp(r1t) and exp(r2t)
2) exp(r1t) and t*exp(r2t)
3) exp(a)(cos(bt) + sin(bt))
What about higher order linear and homogeneous differential equations with constant coefficients? What do we do there?
We still guess y = exp(rt) and find the solutions of the polynomial equation on the variable r.
Then, we write the general solution of the differential equation based on the type of solutions.
What are linear independent solutions of a differential equation?
Those are solutions that, when written as linear combinations equal to zero, all the constants involved must be zero
What is the Wronskian?
It’s an operator based on the determinant that contains the solutions in the first row ans their derivatives in the next ones.
It helps determining if the solutions are linearly independent if the Wronskian is different than zero.
Consider a linear ode of order 2. What should be the condition for satisfying the existence and uniqueness theorem?
All of the function coefficients and f(x) must be continuous in an interval that contains the points involved in the IVP.
General superposition theorem?
If we have n solutions to a nth order linear ode, then the linear combination of them is a solution too.
Moreover, if the solutions are linearly independent, then the linear combination of them is a general solution.
What should we do to solve a 2nd order linear non homogeneous differential equation?
We have to find a solution to the homogeneous equation and another one that’s a particular solution to the non-homogeneous.
(yp + yh) is a solution to the differential equation
How do we solve the 2nd order non-homogeneous linear differential equations with constant coefficients?
We use the method of undetermined coefficients. Given f(x) from the DE, the guess must be y = Af(x)
Undetermined coefficient for exponentials?
A*exp(rt)
Undetermined coefficient for sin(rt) or cos(rt)?
Asin(rt) + Bcos(rt)
Undetermined coefficient for polynomials of degree n?
A0 +A1t + ••• + Ant^n
