Differential Equations Flashcards
Define differential equation
A differential equation is an algebraic equation involving the derivative (or multiple, or higher order derivatives) of a function
How can you write basically any physical process as a differential equation
d/dt(thing) = all effects that increase - all effects that decrease
Separable 1st order ODEs
Called separable if we can write dy/dx = A(x)B(y)
Must check if B(y) = 0 for this
Define unstable equilibrium
Concave down (potential) Derivative is 0
Define stable equilibrium
Concave up (potential) Derivative is 0
Constant coefficient case ODE
y”+py’+qy=0
Assume y=Ce^λx
λ^2+pλ+q=0 after simplification
3 cases - sqrt(p^2-4q)
Properties of complex numbers
z = a+ib
Basically treat it as a vector, it also FOILs, use conjugate when squaring it
Also you can graph it
Euler’s equation
e^iy = cosy + isiny
Constant coefficient homogenous ODE - p^2-4q = +
Solution is y(x) = C1e^λ1x + C2e^λ2x
λ1,2 found from quadratic formula
Constant coefficient homogenous ODE - p^2-4q = -
Solution is y(x) = C1e^[(α+iβ)x] + C2e^[(α-iβ)x]
α = p/2
β = sqrt(p^2-4q)
y(x) = e^αx[Acos(βx) + Bsin(βx)]
Constant coefficient homogenous ODE - p^2-4q = 0
y = C1e^mx + C2xe^mx m = the root = -p/2
The Wronskian
W(y1,y2)(x) = y1(x)y2'(x) - y2(x)y1'(x) W(x) = Waexp(-S(x,a)p(s)ds) where Wa=W(a)
Uniformity of the Wronskian
If y1(x) and y2(x) are any two solutions to the linear equation y"(x) + p(x)y'(x) + q(x)y(x) = 0 On the interval xE[a,b], then their wronskian W[y1,y2](x) is either zero everywhere or zero nowhere on [a,b]
Define qualitative analysis
Learning about the solution to a DE without actually solving it
Direction fields
Tells you what typical solutions look like
DEs tell you the slope of the tangent line at each point, so if you draw it at a bunch of points you can see how a solution looks given a starting point
Define potential
dy/dx = f(y) = -U’(y)
U(y) is called the potential
Numerical approximation methods
Can’t always solve analytically
So we just approximate lol
Forward Euler method
x(i+1) = xi + Δtf(xi,ti)
ti = t0 + nΔt
Comes from first principles
Define discririzing the domain
When you break up the domain into a bunch of tiny pieces
Series solutions
Describing ODEs as power series
Domain is set of x values where series converges
Differentiation and integration is done term by term
And use Taylor’s formula
Define a series being analytic
If series converges in neighbourhood of x0(x0-δ,x0+δ) we say it’s anakytic at x=x0
