Week 3 1st ODE Flashcards
Classify this differential equation (1 step)
ODE
Classify this differential equation (1 step)
ODE
Classify this differential equation (1 step)
ODE
Define PDE
(what does it stand for?)
Partial differential equation
involves partial derivatives of one or more dependent variables with respect to more than one independent variable
Classify this differential equation (1 step)
PDE
Classify this differential equation (1 step)
PDE
Classify this differential equation (1 step)
PDE
What does the order of a differential equation mean?
the order of the highest derivative in the equation
e.g. (d^3x) / (dy^3),
order=3
what’s the order of the differential equation?
order = 1
because of the dy/dx
Which is the dependent and independent variable?
And what do they mean?
dx = dependent variable
what we differentiate of derivative
dt = independent variable
the one we differentiate with respect to
Bracket form:
dependent (independent)
what conditions do linear ODE’s have to satisfy?
Classify this differential equation (2 step)
What is standard form?
- all terms involving dependent variable (TOP) itself/derivative put on LHS
- all terms including independent (BOTTOM)/ constants/ (if not put 0) on RHS
What kind of ODE’s can be homogenous/ non-homogenous?
Only linear ODE’s
how do you distinguish homogenous/ non-homogenous equations?
-Write linear equations in standard form: if RHS= 0, then equation is homogenous
if RHS = non-zero, it’s non-homogenous
What are the solution methods to solve ODE (+ wk4)
1) direct integration
2) Separable variable
3) exact ODE
4) Reducible to separable form
5) Integration factor
solution method: direct integration
what form should the ODE be in to use this method?
Solve this ODE
(what method should you use?)
1) times dx across
2) integrate both sides
Solve this ODE
(what method should you use?)
1) times dx across
2) integrate both sides
solution method: separation variable
what form should the ODE be in to use this method?
The variable y on the RHS prevents using solution method: direct integration
Solve this ODE
(what method should you use?)
Separation variable
-pool y in LHS, pool x in RHS
1) times bracket with y to LHS
2) times dx to RHS
3) integrate on both sides
4) leave +c on RHS
5) express y explicity, solve for y to get generic solution
Solve this ODE
(what method should you use?)
Separation variable
1) factorise y on RHS
2) times dx to RHS,
divide x to RHS
divide y to LHS
3) integrate on both sides
4) keep +c on RHS
5) solve for y and simplify
Solve this ODE
(what method should you use?)
Direct integration
1) move all non-y terms to RHS (including 1)
2) times dx RHS
3) integrate both sides
4) add + c on RHS
What are the 2 types of problem to solve with ODE’s?
General method to solve initial value problem (initial conditions)
1) gives initial condition/ value
2) therefore can find ‘c’ after using a solution method
3) obtain full solution
Solve this to obtain the particular solution
Solve this to obtain the particular solution
End equation gives:
1) particular solution to the ODE alone
2) The solution to the initial value problem
What is the total derivative definition?
What is the requirement for an EXACT ODE
solution method: Exact ODE
what form should the ODE be in to use this method?
Solve this ODE
State what solution method you will use
Exact ODE
solution method: reducible to separable form
what form should the ODE be in to use this method?
Might not be separable, but can be transformed into separable form
is this form separable or not?
Separable
is this form separable or not?
non
is this form separable or not?
Separable
is this form separable or not?
non
is this form separable or not?
Separable
is this form separable or not?
Separable
is this form separable or not?
non
Q1
Solve this ODE
State what solution method you will use
Q2
Solve this ODE
State what solution method you will use
Exact ODE
is this an exact ODE?
Not exact, sign of the number does matter
