Final Exam Flashcards
Ordinary differential equation (ODE)
unknown function x has one independent variable t. Equation relates x and its derivatives to values of t.
First order linear ODE
How to solve?
integrate both sides n-times, will have n arbitrary constants
How to solve?
separation of variables
How to solve?
first-order linear ODE (not separable)
Example: Assume that a drug is absorbed by the body at a rate proportional to the amount of the drug present in the bloodstream at time 𝑡. Let’s write 𝑥(𝑡) for the mg of drug present in the bloodstream at time 𝑡.
Find an ODE for which 𝑥(𝑡) is a solution, assuming that the drug is absorbed at a rate of 0.4𝑥(𝑡) per hour and that the patient is simultaneously receiving the drug intravenously at the constant rate of 10 mg/hour:
N-th order ODE is linear
functions in front of the derivatives only depend on t
N-th order ODE is HOMOGENOUS
E(t) = 0
N-th order ODE is NORMAL
leading coefficient not equal 0
continuous coefficients
General solution to a linear ODE
x = p(t) + H(t)
What is the Existence & uniqueness theorem for ODEs, applies to initial value problems
Operator
What does it mean that linear differential operators are linear?
- L[x1 + x2] = Lx1 + Lx2
- L[kx] = k Lx
Cramer’s rule to solve Bx = A
Replace the ith column of B with the vector A, take determinant.
Linear combination of homogenous solution L[x] = 0
Also a homogenous solution: L[c1h1(t) + c2h2(t)] = 0
If you found n linearly independent solutions to a n’th order ode, then any other solution can be written as a linear combination of those
Existence & uniqueness theorem for nth order linear ODEs
If you have n initial conditions, then x = x(t) has a UNIQUE solution for a fixed value t0.
Linearly dependent vs independent
Wornskian test for linear independence:
Theorem: Wronskian for solutions, non-vanishing never vanishing:
Exponential shift formula
Euler’s formula
How to solve a homogeneous linear equation with constant coefficients:
Find a particular solution p(t) to linear ODE, P(D)x = E(t)
Find equivalent system of ODEs in matrix form: Dx = Ax + E