Study Sheet: Solving & Analyzing First Order Equations & Systems, Lecture notes of Mathematics

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Differential Equations Study Sheet

Matthew Chesnes

It’s all about the Mathematics!

Kenyon College

Exam date: May 11, 2000

6:30 P.M.

1 First Order Differential Equations

• Differential equations can be used to explain and predict new facts for about everything that changes continuously.

d^2 x dt^2

• a

dx dt

• kx = 0.

• t is the independent variable, x is the dependent variable, a and k are parameters.
• The order of a differential equation is the highest deriviative in the equation.
• A differential equation is linear if it is linear in parameters such that the coefficients on each deriviative of y term is a function of the independent variable (t).
• Solutions: Explicit → Written as a function of the independent variable. Implicit → Written as a function of both y and t. (defines one or more explicit solutions.

1.1 Population Model

• Model: dP dt = kP.
• Equilibrium solution occurs when dP dt

• Solution: P (t) = Ae(kt).
• If k > 0, then limt→∞ P (t) = ∞. If k < 0, then limt→∞ P (t) = 0.
• Redefine model so it doesn’t blow up to infinity.
• dP dt = kP (1 −

P

N

• N is the carrying capacity of the population.

1.2 Seperation of Variables Technique

dy dt = g(t)h(y).

h(y) dy = g(t)dt.

• Integrate both sides and solve for y.
• You might lose the solution h(y) = 0.

1.7 Bifurcations

• Bifurcations occur at parameters where the equilibrium profile changes.
• Draw phase lines (y) for several values of a.

1.8 Linear Differential Equations and Integrating Factors

• Properties of Linear DE: If yp and yh are both solutions to a differential equation, (particular and homogeneous), then yp + yh is also a solution.
• Using the integrating factor to solve linear differential equations such that dy dt

+P (t)y = f (t).

• The integrating factor is therefore e(

∫ (^) P (t)dt) .

• Multiply both sides by the integrating factor.
• e(

∫ (^) P (t)dt) dy dt

• e(

∫ (^) P (t)dt) P (t)y = e(

∫ (^) P (t)dt) f (t).

• then via chain rule ...
• d{e(

∫ (^) P (t)dt) y} dt

((Integrating factor * y))= e(

∫ (^) P (t)dt) f (t).

• Then integrate to find solution.

1.9 Integration by Parts

udv = uv −

vdu.

2 Systems

dx dt = ax − bxy,

dy dt = −cy + dxy.

• Equilibrium occurs when both differential equations are equal to zero.
• a and c are growth effects and b and d are interaction effects.
• To verify that x(t), y(t) is a solution to a system, take the deriviative of each and compare them to the originial differerential equations with x and y plugged in.
• Converting a second order differential equation, d^2 y dt^2 = y. Let v = dy dt . Thus dv = d^2 y dt

2.1 Vector Notation

• A system of the form dx dt = ax + bxy and dy dt = cy + exy can be written in vector notation.
• d dt P(t) =

dx dt dy dt

[

ax + bxy cy + exy

]

2.2 Decoupled System

• Completely decoupled: dx dt

= f (x), dy dt

= g(y).

• Partially decoupled: dx dt = f (x), dy dt = g(x, y).

3.2 Stability

Consider a linear 2 dimensional system with two nonzero, real, distinct eigenvalues, λ 1 and λ 2.

• If both eigenvalues are positive then the origin is a source (unstable).
• If both eigenvalues are negative then the origin is a sink (stable).
• If the eigenvalues have different signs, then the origin is a saddle (unstable).

3.3 Complex Eigenvalues

• Euler’s Formula: ea+ib^ = eaeib = eacos(b) + ieasin(b).
• Given real and complex parts of a solution, the two parts can be treated as seperate independent solutions and used in the linearization theorem to determine the general solution.
• Stability: consider a linear two dimensional system with complex eigenvalues λ 1 = a+ib and λ 2 = a − ib. - If a is negative then solution spiral towards the origin (spiral sink). - If a is positive then the solutions spiral away from the origin (spiral source). - If a = 0 the solutions are periodic closed paths (neutral centers).

3.4 Repeated Eigenvalues

• Given the system, d dt X = AX with one repeated eigenvalue, λ 1.
• If V1 is an eigenvector, then X 1 (t) = eλtV 1 is a straight line solution.
• Another solution is of the form X 2 (t) = eλt(tV 1 + V 2 ).
• Where V 1 = (A − λI)V 2.
• X 1 and X 2 will be independent and the general solution is formed in the usual manner.

3.5 Zero as an Eigenvalue

• If zero is an eigenvector, nothing changes but the form of the general solution is now

X(t) = k 1 V 1 + k 2 eλ^2 tV 2.

4 Second Order Differential Equations

• Form: d^2 y dt^2 + p(t) dy dt = q(t)y = f (t).
• Homogeneous if f (t) = 0.
• given solutions y 1 and y 2 to the 2nd order differential equation, you must check the Wronskian if both solutions are from real roots of the characteristic.
• W = det

[

y 1 y 2 y′ 1 y 2 ′

]

• If W is equal to 0 anywhere on the interval of consideration, then y 1 and y 2 are not linearly independent.
• General solution given y 1 and y 2 is found as usual by the linearization theorem.
• Characteristic polynomial of a 2nd order with constant coefficients: as^2 + bs + c = 0.
• Solutions of the form y(t) = est.
• s = − b 2 a

b^2 − 4 ac 2 a

• if b^2 − 4 ac > 0, then two distinct real roots.
• if b^2 − 4 ac < 0, then complex roots.
• b^2 − 4 ac = 0, then repeated real roots.

4.1 Two real distinct Roots

• Two real roots, s 1 and s 2.
• General solution = y(t) = k 1 es^1 t^ + k 2 es−^2 t.

4.2 Complex Roots

• Complex Roots, s 1 = p + iq and s 2 = p − iq.
• General solution = y(t) = k 1 eptcos(qt) + k 2 eptsin(qt).

4.3 Repeated Roots

• Repeated Root, s 1.
• General solution = y(t) = k 1 e

− b 2 a t

• k 2 te

− b a 2 t .

5 LaPlace Transformations

• Definition L{f (t)} =

0 e −stf (t)dt = limT →∞^ ∫^ T 0 e −stf (t)dt.

• ONLY PROVIDED THAT THE INTEGRAL CONVERGES!!! MUST BE OF EX- PONENTIAL ORDER!!!
• L{f (t)} = F (s).
• L{ 1 } =

s

• L{t} =

s^2

• L{eat} =

s − a

• L{sin(ωt)} =

ω s^2 + ω^2

• L{cos(ωt)} = s s^2 + ω^2

• Linear: L{αf (t) + βg(t)} = αF (s) + βG(s).

5.1 Inverse Laplace Transforms

• Linear: L−^1 {αF (s) + βG(s)} = αf (t) + βg(t).

5.2 Transform of a derivative

• L{f ′(t)} = sL(f (t) − f (0).
• L{f ′′(t)} = s^2 L(f (t) − sf (0) − f ′(0).