Modelling with Differential Equations Functionin x variable

Differential Equation Efaation involving yt andits derivatives

y y

EXT may also involve se

y y y sty od

sincy date yOrder at differential highest derivative in

equation equation1st order 2nd OrderExamples

L

y y y Hey 0

Important f ex is a solution to a givendifferential equation it y tix

E sSatisfies equation

y yobserve e e y e is a solution

Important Remara of y y

The equality must be an equality of Functions

For example if that is a solution to y sexythen 1 x x f Csc for all K

PrimaryEoat find all possible solutions to a

given differential equation

Special case

y goaly text a solution F Cal

g ca

7 an anti derivative 012Conclusion

solving y gcx Calculating JgcsaidxExample

Jaday x

any constantL7 x x text text 1

Observation A given differential equation does not

have a single solution but a family of solutions

Other Examples any constant

rY l

y y y Ae are solutionsan constants

7 y y y Acosta Bsiutx are solutions

W e

Differential equations are everywhere They modelcountless real world problems

Basic IdeaMathematicalModel

Real World Gweening DifferentialProblem Equation

Mathematical Real WorldPv dich onsconclusion Interpret

E Motion of a Spring

I 1 Newtons 2ndLaw

33

ofF

E IW

tI Hooke's Lawmass in Force kggut u I

positive constantCSpring constant

2nd Ordermy kg y y DAT Eg

Special Cau i k I m Inot obvious this is all solutions

y y y A coset Dsin Ct

Remain We need more information to get specificA and B Called initial

conditionsExample y lol I y co 0

y coA s o t D Siu O I

c y co AA sin o t B s o o B O

yet coset c Oscillating motion as

expected

Remark To get a single solution to a differential

solution initial conditions must be givenFixed numbersI1st order y Coco To

2nd Ordery Lao yo y LX o l y

nth order Ch 1

y Cao yo i y Xo Ju i