week 2 - preliminary concepts Flashcards
what does this symbol mean ẋ
it is a function of f(x,t) where it is a function of both x and time where
x : I ⊆ R → R; t → x(t)
f : I × R → R; (t, x) → f(t, x)
the unknown is not a number but a function x=x(t)
when is a differential equation first order?
it is first order if only ẋ and not higher order derivatives (e.g. ẍ)
when is a differential equation scalar?
it is scalar when x(t) is one dimensional (not a matrix)
when is a differential equation ordinary?
it is ordinary if no partial derivatives are present
what is the definition of an autonomous ODEs?
An ordinary differential equation is autonomous if the function f does
not directly depend on t:
ẋ = f(x)
example ẋ = e^x is autonomous while ẋ=te^x is not as it depends on x and t
what is the defintion of a particular solution?
Consider a continuous and differentiable real value function φ(t), with
t ∈ J ⊆ I. We say x = φ(t) is a particular solution of the ODE
ẋ=f(x,t) on the interval J, if
φ^(.) = f(φ, t) ∀t ∈ J
what is the definition of a general solution?
the set of all particular solutions is called the general solution, it is a family of functions. the objective of solving an ODE is to find its general solution
what is the definition of an integral curve?
the graph of a particular solution is called an integral curve
what is the definition of an intial value problem?
an initial value problem is an ODE with the value of x at time t0 ∈ I exogenously given:
ẋ = f(x, t) with x(t0) = x
what is the definition of seperability?
a scalar ODE ẋ=f(x,t) is seperable if it can be rewritten as ẋ =g(t)*h(x)
what is the method for solving seperable ODE?
1) check whether the ODE is seperable
2) remember that ẋ =dx/dt we have that dx/dt =g(t)*h(x) is equivelent to dx/h(x)=g(t)dt
3) intergrate both sides and then solve for x to find the solution of the ODE
