Download Classification of Differentiation Equations - Review Sheet | MATH 305 and more Study notes Mathematics in PDF only on Docsity! K.A. Pericak-Spector Math 305 Classification Spring 2005 These are the basic notes from the first day of class. Differential equations are important in many aspects of engineering, science, economics, and everyday life. Anytime a quantity is changing with respect to time or position a differential equation could be involved. For example, how the population of fruit flies in your kitchen is increasing your speed as you came to school this morning how much interest your savings account is making how the strength of your coffee changes as the water filters down through the grounds. DEFINITION. A differential equation (DE) is an equation involving an unknown function and its derivatives. If the unknown function depends on a single independent variable ))(( xy then only ordinary derivatives appear in the DE and we say it is an ordinary differential equation (ODE). If the unknown function depends on more than one independent variable )),(( yxu then partial derivatives will appear and we say it is a partial differential equation (PDE). EXAMPLES. A. mass spring 0=+′+′′ kxxcxm x… is the position of the object from equilibrium at any time t m…is the mass of the object c… is the damping or viscosity k… is the spring constant B. population PbPadtdP )1(/ −= P… population at any time t a,b…are positive constants and represent birth and death rates respectively Notice that if bP /1= then 0/ =dtdP and the population is not changing. If 01 <− bP then dtdP / is negative and P is decreasing. If 01 >− bP then dtdP / is positive and P is increasing. These are examples of ODEs. Some examples of PDEs: C. potential 0=+ yyxx uu u… potential at any x and y D. diffusion or heat xxt uu 2α= u… heat in a 1-dimensional bar at any x and t α …material constant If derivatives are denoted by primes )(′ , dots )( . , or dx d , the equation must be an ODE. If derivatives are denoted by subscripts )( x or y∂ ∂ the equation must be a PDE. REMARK. nnn dxydy /)( = Math 305 Classification Page 2 State whether the following are ODE or PDEs also determine the independent variable and the dependent variable. 1. 03 =+′′+′′′ xxtx 2. 04 =+′′ xx 3. 0)4( =+ xx 4. 0sin2 2 =+ tt dt xdm 5. 0=++ ttxxyy uuu 6. 02 2 2 2 2 = ∂ ∂ − ∂ ∂ x uc t u 7. 0=+′ xyy 8. 0=+ xt uuu What we want to be able to do is given a DE we would like to be able to solve for the unknown function. DEFINITION. The order of a DE is the highest number of derivatives that appear in any one term in the equation. In the above examples we see that (A) is 2nd order, (B) is 1st order, (C) is 2nd order, and (D) is 2nd order. Determine the order in problems 1-8. We will usually write ),...,,( )1()( −′= nn yyyxfy where f is some known function of it arguments. DEFINITION. A solution of the ODE ( ))1()( ,...,, −′= nn yyyxfy of the interval βα << x is the function ϕ such that )(,..., nϕϕϕ ′′′ exist and satisfy ( ))1()( ,...,,, −′= nn xf ϕϕϕϕ for all ),( βα∈x .
