Math 251, Exam 1 Review

What are the steps to solving a separable differential equation?

1. Rearrange the equation so all variables are on their own side 2. Integrate both sides with respect to their variables, remembering the constant on one side 3. Solve for C, usually with initial conditions 4. Try to find an explicit solution if possible

Linear Differential Equation

A differential equation in which the dependent variable (or "y" in the case of "dy/dx")

Partial Differential Equation

A differential equation involving a function of 2 or more variables AND their partial derivatives (sample ex in picture)

Ordinary Differential Equation

A differential equation involving a function of only 1 variable

Autonomous Equation

A differential equation where the independent variable does not explicitly appear in its expression (only y's). If it's autonomous, it MUST be separable

Steady state

All non-transient terms

Equilibrium Solutions

Also known as constant solutions

Constant solutions

Also known as equilibrium solutions or stationary points. First order functions become constant numbers, and accordingly, their derivatives become zeros. A "solution" is the result after the simplification

Differential Equation (ex?)

An equation that contains one or more derivatives of an unknown function (sample exs: y'' - 3y' + 5y = t^2 or y''(t^2 • y') = (ye^(5t))/t

Transient term

Any term in the solution y(t) whose limit is 0 as t approaches infinity

How do you put a formula into standard form?

For a first-order, linear differential equation, make sure the coefficient in front of y' is 1. It will leave you with the form [ y' + p(t)y = g(t) ]

What are the steps for integrating factor method?

Sample: [ y' + p(t)y = g(t) ] 1. The equation has to be placed in standard form 2. Find integrating factor [ u(t) = e^(integral(p(t)dt))] 3. Find the general solution by multiplying both sides by IF 4. Use initial value to solve for C

What makes an equation "separable"?

The equation is separable if it can be written in the form [ M(x) + N(y)y' = 0 ], where m(x) contains only x's and N(y) contains only y's. It allows the x's and y's to be separated to different sides of the equation

Order

The highest derivative taken in a differential equation (note: NOT powers)

What is our differential form of the population model?

dP/dt = kP(M - P) P(0) = initial population M= limiting population/carrying capacity Solution of P(t) = [MP(0)e^(kMt)] / [M +P(0)(e^(kMt) - 1)