Differential Equations Test #1

What form must you have to use the method of integrating factors?

y’ + p(t)y = g(t)

What is the integrating factor formula?

u(t) = e^ integral of p(t)dt

What is the final full formula to find y(t) with the method of integrating the factor?

integral of (u(t)g(t))/ u(t)

If your finding the limit for a function as it approaches a specific value…

Just plug the value into the equation

Before applying initial conditions, C is?

An arbitrary constant

After applying initial conditions, C is?

A specific value to be solved for.

If you use algebra to get a square root- your answer must have..

a plus or minus ±

When trying to find if equilibria are stable or unstable?

Test a number from each interval into the Diff Eq and see if the result is positive or negative

If in a given interval on a diff eq, the answer is positive…

slopes are positive and will be increasing for all of time

If on a given interval of a diff eq, the answer is negative….

the slope is negative and will be decreasing on the interval for all of time.

If you have a quadratic in your solution to separation of variables DQ…

± sqrt() is the technical answer, but take whichever one the initial function pivots you to, ex y(0)= -2 then chose the - branch.

When solving 1st order autonomous equations, and its asking for a limit as t approaches infinity for a certain value of t…

look at the little chart you’ve made after finding the eq solutions, and decide which, if any, eq solution that starting value will approach after a while. (if its below 2 and increasing there, the limit is 2 as t—> infinity).

What does the logisitic equation model?

Population growth with a limited carrying capacity

Logistic Eqn?

(r-ay)y= y’

Carrying capacity k=

r/a

Explicit solution to the Logistic Equation?

y(t) = k/ (1+ Ce^(-rt))

The logistic eqn is another example of…

a first order autonomous eqn, can be solved with algebra and set dy/dt= 0, test intervals with values between equilibria to find increasing and decreasing.

Newtons Law of Cooling Eqn

T(t) = (To-Ta)e^(-kt) + Ta

Ta in the newtons law of cooling eqn is…

the temp of the surroundings

To in the newtons law of cooling is…

the initial temp

Euler’s Step Formula

y= y^o + f(x^o + y^o)* h

h in euler’s formula

x initial- x final over # of steps

x1 =

Xo + h

EAUT for a first order linear equation…

There exists a unique solution through the interval condition as long as p(x) and g(x) are continuous on an interval containing the inital point.

Discontinuities that will ‘break’ the EAUT…

denominator is zero, square root of a negative, log of zero or negative, tangent/secant where cos(x)=0

The derivative of sinx is ____ but the integral of sinx is ____

cosx, -cosx

The derivative of cosx is ____ but the integral of cosx is _____

-sinx, sinx