Differential Equations Pictures - Differential Equations with Linear Algebra | MAT 2705, Study notes of Differential Equations

Download Differential Equations Pictures - Differential Equations with Linear Algebra | MAT 2705 and more Study notes Differential Equations in PDF only on Docsity! DE pictures: sinusoidal example uginuseidal funchdn" ywrg sO oa Soluhéns : ye GX + Cz SInX = A cos(e 8) ornplitude. 4 + unit Frequency Phase shifr & Cr) “sinusoidal furctions"ace prase shifted Cosines OF Sines with an amolil ice ASA ~& y= Acosx Y= Acos(x-8) weed S70 as shown: some ampthude peaks occurat afee phase later value of x same phase shift a Fferent amplitude " phase“ is just ancther word for the input angle for trig function: Cost...) prose <j (aC2): SOLN SPACE PICTURE C. CO), C2) w rsx + CaS = A cos(x-8) stax ~ ©) q Chi) ~~ 00s x, LYru2t = Losx 1 Stnx} basis f selnspace corresponds tp basts {7 =Ci0), J=(o, Yt of coefficient plane, each soln corresponds toa 2-veckor, (Enz) Wee Cartesian coords (wy), (A, 5) like polar coords (ry 6), [detwils Ller} Theunitcircle At*~ c24c = 4 corresponds ty all usiek ampliade Sduhons: ») tos(x-8) ~COSKe 56 us S>o q ‘ d<o N\-sinx = cs(xete) Pothng solns inthe sein space pichiwe visually shaus relative amplihideand phrase which characlenzes what you see ta the oscilloscope picture, Ce Az i CY Note that if xis a time vanable, the curve with peaks at larger x values is behind ih time (later time) $2-§1 7 © as shown >| assnoun cy Th this sense the sine lags behind the cosine by 90°: S=ty2 30.