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(MATH150)2000_and_2001_spring_midterm_exam.pdf
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Math150,MidtermExam,Spring2000

1.SolvethefollowingBernoulliequation(20pts)
1
022
y;y.2ytant.t.0: t
2.Sketchthesolutionsofthefollowingequationandindicatestableandunstable
equilibriumsolutions(qualitativeanalysis)(20pts):
dy
.;(y;1)(y;2)2(y;3): dt
3.Solvethefollowingproblembythemethodofexactequationsandintegrating
factors(20pts):
ydx+(2xy 2+2y 3+4y+x)dy.0.y(0).2:
4.Findthesolutionofthefollowinginitialvalueproblem(20pts):
dyx 2+3y 2 ..y(1).;2: dx2xy
5.Findthesolutionofthefollowinginitialvalueproblem(20pts):
000
y;4y;5y.0.y(0).1.y0(0).1:
1
Math150,MidtermExam,Spring2001

1.SolvethefollowingBernoulliequation(7pts)
1
02
y+y.ysinlnt.t.0: t
2.Findthevaluesofaandbsuchthatthefollowingdi.erentialequationisexact. Usingthesevaluesofaandbtosolvetheequation(7pts):
xx
(aesiny;2ysinx+2cos2x)dx+(ecosy+bcosx;3y 2)dy.0.y(0).2:
3.Findthesolutionofthefollowingdi.erentialequationbythemethodofundeter-minedcoe.cients(8pts)
000ty;2y+y.te+4.y(0).1.y0(0).0:
4.Solvethefollowingdi.erentialequationbythemethodofvariationofparameters (8pts):
.
00
y+4y.4cot2t.0.t.: 4
2