Classifying Differential Equations: A Comprehensive Guide
School
University of Illinois, Urbana Champaign**We aren't endorsed by this school
Course
MATH 285
Subject
Mathematics
Date
Dec 11, 2024
Pages
2
Uploaded by UltraResolve11737
Name:Homework 1, Due: Aug. 28, 2023Math 285 B1/C1/F1Answer:Problem 1 [4pts]Classify the following equations as linear or nonlin-ear, ordinary or partial, and give the order.@u@t=@2u@x2d3ydx3+ 17ydydx-11 = 0etd4ydt4+t6dydt-t27y= cos(t)ut+uxxx+uux= 0Answer:Problem 2 [3pts]For each of the following equations find a solutionof the indicated formd2ydt2+ 3dydt+ 2y= 0y=ertd2ydt2-5dydt+ 4y= 0y=ertt2d2ydt2+tdydt-4y= 0y=trt2d2ydt2+ 4tdydt-4y= 0y=trAnswer:Problem 3 [3pts]Verify that the functiony(t) =1cos(t)is a solution to the di↵erentialequationd2ydt2= 2y3-yIMPORTANT PLEASE READ:Your homework should be uploaded to the course Canvas space.Please make sure that your upload is no more than a couple of meg at most or it will crash the gradingprogram and you will not get credit for the homework.Please make sure that the first page of your upload is this sheet, with the answers filled in boxes above.You may attach as many additional sheets as is necessary to show your work.Failure to follow these instructions will result in your not receiving credit for this assign-ment.1ShivanshAgarwalLinear,partial.OrderanNon-Linear,ordinary,3rdorderLinearordinary,LYMorderNon-Linear,partial,23rdorder(workingisonthenextpage]1.y=e-t,ye-2t1.2.y=et,yet2.3.G:t2,y=t-23.Y.y=t',y=t4.-(st):(sect)=sect.tant=(sect,tantsect.tan't+sectsimplifiedas-sect+2secttttst-ayo-yHenified
PROBLEM-21.y=ert+3+2y=0firstderivative-by:rett3substituting:rert+3vert+Cert=0secondderivative-II:vertert(r3r2)=0ert0or2+3r2=0notpossiblerGr+u20=r(r+2)+1(r+2)0..r=-1or-22.yert454y=0firstderivative-by:rett3substituting:vert-swert+yert:osecondderivative-II:vertert(r2-5r+4)0ertsibleorr-3r+40=r-yr-r+4=0notpor(ry)1(r4)07..r1orY3.y=tr+x2++dy-4y=firstderivative->Crtr-3ingtsubstitutitr(r-1)+r-+trt-1-4t=0Er(r1)++tr-4t0secondderivative-Ir(r-1e12//r(r-1)++r.tr-4t=0=(r2-rr-4)t0->r2y0ort0notpossible:.12Y.y=tr+2+4t4y=0firstderivative->Crtr-3t2(02-r)+r4A.r+-4t0//secondderivative-Iv(r-er-2substituting:/+2tt⊥notpossibleEtr(r-rYr4)=0=t0or+3r-4=022+Yrr-y=0=7r(r+y)1(r+y)=0.r=10rr=4