Transcription of MA214 Fall 2008 Practice Test #2 - University of …
1 MA214 fall 2008 Practice Test #2 MATERIAL: Chapter 3, sections , , , , ; Chapter 6, sections We ll do the word problems of sections and section after theThanksgiving break so this material will NOT be on test Show thaty1(x) =xandy2(x) =xexform an independent set of solutionsto the ODE:x2y x(x+ 2)y + (x+ 2)y= 0,x> Are the two functionsw1(t) =t(t+ 1) andw2(t) =t2independent on thereal line?3. Find the most general solution to the ODE:y (t) ty (t) =t,using the substitutionu(t) =y (t).4. Find the unique solution to the initial value problemy + 5y 6y= 0,with the conditionsy(0) = 1 andy (0) = Compute the Laplace transform off(t) = Use Abel s formula to compute the Wronskian of two solutions to the ODE2t2y + 3ty y= 0,t> Use variation of parameters to find a particular solution toy + 4y + 4y=t 2e Find the Laplace transform of the following functions:f(t) =tsin 4t(use integration by parts)g(t) = 00 t 1(t 1)t 1 21t 29.
2 Find the most general real solution to the ODE:y + 2y + 2y= Find the unique solution to the initial value problem4y + 12y + 9y= 0,withy(0) = 1, andy (0) = Find the unique solution to the initial value problemy + 4y= 3 sin 2t,withy(0) = 2 andy (0) = Consider a driven, undamped harmonic oscillator described by the ODEu + 2u= 2 cos is the natural frequency? Find the solutions for not equal to,and for equal to, the natural frequency, when the initial conditions areu(0) = 0 andu (0) = What is a set of independentrealsolutions for the damped, undrivenoscillator described byu + 4u + 4u= is the unique solution to this ODE with initial conditionsu(0) = 1andu (0) = 0? How long does it take for the amplitude to decrease toone-half of its initial value?
3 14. Use the Laplace transform method to solve the initial value problemy 2y 2y=u2(t),y(0) = 2,y (0) = Use the method of undetermined coefficients to find the unique solutionofy + 4y=t,y(0) = 1,y (0) = Find the inverse Laplace transform of the functions:F(s) =e ss(s2+ 1),F(s) =2s 3s2+ 2s+
