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MTH 205-Pratice problems and solutions for Test -02-Su-09

# MTH 205-Pratice problems and solutions for Test -02-Su-09 -...

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Unformatted text preview: M'I'HZEIS: Practice Problems Exam 2 i Summero‘j Ql. Find an interval around x = 0 for which the initial value problem (x—I)y"+4_‘x,y'+y=sin(x). y(0)=l. y'(0)=0 has a unique solution. QZ. Consider the differential equation yr! _ 4y = 82: a) Find the general solution y c of the corresponding homogeneous eguation. b) Find a particular solution y p by the method of undetermined coefficients. c) Find the general solution. d) Determine the solution subject to the initial conditions: y(0) = 2. y'IO) =1 I Q3. Solve the same problem in Q3. by the method of variation of parameters. Q4. Find the general solution of .Iczja"'-41I:y'-I-6y=.1cJ QS. Given that y. =e‘ is a solution ofthe d. e (xi-Dy" —(x+ 2)y + y= 0. find the general solution. Q6. Set up the appropriate form of the particular solution yp , but do not determine the values of the coefﬁcients. ym +9y' = (x + l) sin(3x) Q7. Use the method of undetermined coefﬁcients to ﬁnd the solution of the I. V P. y” + 4y= 25in(2t), y(0)— _ I, y (0): 0 08. Given that { y, =x , 2 — -x lnx} Isa fundamental set of solutions of the corresponding homogeneous equation of the differential equation xy" -'3:I.y +4 y x lnx Use the variational method to find the general solution of the differential equatiOn. Q9. A 16 — pound weight stretches a spring 2 feet. Initially the weightstarts from rest 2 feet below the equilibrium position. Determine the differential equation and the initial conditions of the motion, if the surrounding medium offers aidamping force numerically equal to the instantaneous velocity and the weight is driven by an external force equal to f (r) = S 005(21). 010. 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