Final Exam Differential Equations

Complex roots with a positive real part…

Spiral outward

Complex roots with a negative real part…

Spiral inward

What is the origin called in the case of complex roots with a real part, a?

Spiral Point

Name of the origin when theres complex roots with no real part, a?

Center

When the roots are real and have opposite signs?

The origin is called a ‘saddle point’ and the real general solution takes the form, x(t)= c1v1e^r1*t+ c2v2e^r2*t

Roots are real and distinct, both have the same sign

Origin is a node

Roots are real and distinct, both negative

Stable node, all solutions move towards the origin

Roots are real and distinct, both are positve

Unstable node, all solutions move away from the origin

Clockwise Rotation Matrix

cos(t) sin(t)

-sin(t) cos(t)

Counterclockwise Rotation Matrix

cos(t) -sin(t)

sin(t) cos(t)

Solution of the form +- ib have

circular

Rotation with an exponential coeff have a… shape

elliptical

Method of integrating the factor FORM?

y’ +p(t)y = q(t)

Method of integrating the factor formula?

u(t) = e^{integral (p(t))}, y(t)= integral (u(t)q(t)dt / u(t)

What can you ABSOLUTELY not forget after method integrating factors after the second integration?

the +C

Population growth/Logistic equation

dy/dt = ( r-ay)y

‘y’ in the popoulation equation is?

popultion at time ‘t’

r in the population equation?

Intrinsic growth rate

‘a’ in the population equation is?

the ‘competition/crowding’ coefficient.

Carrying capacity of the population equation?

K= r/a

Logistic equation is an example of which kind of differential equation?

autonomous, can be solved by looking at the equilibrium solutions: where the equation = 0, and if the slope is positive or negative in the regions between equilibria.

Transient solution, yh

the homogenous solution, the solution that dies off

yp, the steady state solution

the solution from solving the nonhomogenous part, the

the solution to a flow problem where the flow in is equal to the rate of flow out?

Q(t)= EqSolution + Ce^{t*(coefficient of Q(t) in the equation)}

The water tank problems are and example of?

a first order autonomous differential equation, can find equilibrium solutions by setting dQ/dt= 0

e^aln(x)=…

x^a

When you see, ‘growth rate proportional to the population’?

y’= ky, there is no ‘a’ crowding coefficient in a logistic equation.

General solution to ‘growth rate proportional to population’

y(t)= Ce^kt

Laplace of t²

2!/ s³

Laplce of u(t-2)g(t-2)

e^-2s*G(s)

Laplace of the heaviside?

e^-sc/s