Differential Equations Unit 2-4

Bernoulli Equation

First order ODE of the form y’ + py = qy^a, where a =/= 0 or 1

1. Set v = y^1-a

2. Find y and y’ in terms of v

3. Substitute v for y

4. Use integrating factor

5. Integrate

6. Substitute y for v

7. Isolate y

Theory of Existence & Uniqueness for 1st Order IVP

{ y’ + py = q

{ y’ = f(x,y)

{ y(a) = b

If there exists an interval I around x=a where p & q are continuous, then there exists (f is continuous) only one (df/dy is continuous) solution to the IVP over all of I

autonomous equation

RHS not dependent on independent variable that the variable of interest is made derivative WRT

Bifurcation diagram

xy graph where y is population at equilibrium and x is rate of population depletion

Schaefer Model

Ps = 0 & K(1-E/r)

Ym is where E = R/2

equivalent integral equation

Given y’ = f(x,y)

y(x) = int from 0 to x of f(s, y(s)) ds

Method of Successive Approximation / Picard Iteration

y₀(x) = 0

y₁(x) = int 0 to x of f(s, y₀(s))ds

y_n(x) = int 0 to x of f(s, y_n-1(s))ds

Find sigma notation, prove it is successive by induction, and convert into elementary function

Principle of Superposition

Any linear combination of solutions for a linear homogeneous equation is also a solution

Theorem of Existence & Uniqueness for 2nd Order Linear IVP

{ y’’ + py’ + qy = f(x)

{ y(x0) = a

{ y(x0) = b

If p, q, and f are continuous in an interval I around x-, then only one solution exists to the IVP over all of I

equidimensional equation

equation that fits the substitution

y = x^a

y’ = ax^a-1

y” = a(a-1)x^a-2

Euler’s Identities

cos(x) = ½(e^ix + e^-ix)

sin(x) = ½(e^ix - e^-ix/i)

Wronskian

det of y1(x) y2(x)

. y’1(x) y’2(x)

Abel’s Theorem

Given two solutions y₁ and y₂ of linear homogeneous ODE in interval I,

if they are linearly independent, then

W(y₁, y₂)(x) ≠ 0, ∀ x ∈ I 

if they are linearly dependent,

W(y₁, y₂)(x) = 0, ∀ x ∈ I

Solving second order homogeneous IVP

1. Find two linearly independent solutions

2. Verify with Abel’s Theorem

3. Apply ICs

4. Solve for constants

5. Find I, interval around x₀, where p & q are continuous

constant coefficient 2nd order equation

Substitute y = re^rx

repeated roots 2nd order eq

multiply first solution by x to get second solution

complex conjugate root 2nd order eq

1. Use e^ix = cos(x) + i sin(x)

2. Set C3 to C1 + C2 and C4 to i(C1 - C2)