5 can verify that the Wronskian of yıTO OUX (TE)(CE)#0 for t 0.Upenitatt ballConsequently, y, and y2 form a fundamental set of solutions of equation (33) for t 0.Problemsmont bonumineIn each of Problems 1 through 8, find the general solution of the givendifferential equation.100001. y" - 2y + y = 02. 9y" +6y' + y = 03.4y" - 4y' - 3y = 04.y" - 2y +10y = 05.y" - 6y' +9y=06.4y" +17y + 4y = 07. 16y" +24y' +9y = 08. 2y" +2y'+y=0In each of Problems 9 through 11, solve the given initial valueproblem. Sketch the graph of the solution and describe its behaviorfor increasing t.9. 9y" - 12y' + 4y = 0,10. y"-6y' +9y = 0,11. y" + 4y + 4y = 0,y(0) = 2, y'(0) = -1y(0) = 0, y'(0) = 2y(-1) = 2, y'(-1) = 112. Consider the following modification of the initial value problemin Example 2:y" - y' += 0, y(0) = 2, y'(0) = b.naFind the solution as a function of b, and then determine the criticalvalue of b that separates solutions that remain positive for all t 0from those that eventually become negative.3,-3/2W[y₁, y2l(t) = 2The Shosath0=0x+1)+KopbN 13. Consider the initial value problem4y" +4y' + y = 0, y(0) = 1, y'(0) = 2.(36)1600a. Solve the initial value problem and plot the solution.b. Determine the coordinates (tM, YM) of the maximum point.c. Change the second initial condition to y'(0) = b 0 andfind the solution as a function of b.d. Find the coordinates (tM, YM) of the maximum point in termsof b. Describe the dependence of tM and yM on b as b increases.14. Consider the equation ay" + by' + cy= 0. If the roots of thecorresponding characteristic equation are real, show that a solution tothe differential equation either is everywhere zero or else can take onthe value zero at most once.Problems 15 through 17 indicate other ways of finding the secondsolution when the characteristic equation has repeated roots.conIn each oforder to fir15. a. Consider the equation y" + 2ay' + a²y = 0. Show that theroots of the characteristic equation are r₁ r₂ = -a so that onesolution of the equation is eat.b. Use Abel's formula [equation (23) of Section 3.2] to showthat the Wronskian of any two solutions of the given equation isW(t) = y₁(t) y₂(t) - y₁ (1) y₂(t) = c₁e-2at,18. 12y19. Fy20. 1221. xywhere c₁ is a constant.c. Let y₁ (t) =eat and use the result of part b to obtaina differential equation satisfied by a second solution y2(t). Bysolving this equation, show that y2(t) = te-at.22. X23. Tarisescircuand24.equsol

Question

5 can verify that the Wronskian of yıTO OUX (TE)(CE)#0 for t> 0.Upenitatt ballConsequently, y, and y2 form a fundamental set of solutions of equation (33) for t > 0.Problemsmont bonumineIn each of Problems 1 through 8, find the general solution of the givendifferential equation.100001. y&#34; - 2y + y = 02. 9y&#34; +6y' + y = 03.4y&#34; - 4y' - 3y = 04.y&#34; - 2y +10y = 05.y&#34; - 6y' +9y=06.4y&#34; +17y + 4y = 07. 16y&#34; +24y' +9y = 08. 2y&#34; +2y'+y=0In each of Problems 9 through 11, solve the given initial valueproblem. Sketch the graph of the solution and describe its behaviorfor increasing t.9. 9y&#34; - 12y' + 4y = 0,10. y&#34;-6y' +9y = 0,11. y&#34; + 4y + 4y = 0,y(0) = 2, y'(0) = -1y(0) = 0, y'(0) = 2y(-1) = 2, y'(-1) = 112. Consider the following modification of the initial value problemin Example 2:y&#34; - y' += 0, y(0) = 2, y'(0) = b.naFind the solution as a function of b, and then determine the criticalvalue of b that separates solutions that remain positive for all t > 0from those that eventually become negative.3,-3/2W[y₁, y2l(t) = 2The Shosath0=0x+1)+KopbN 13. Consider the initial value problem4y&#34; +4y' + y = 0, y(0) = 1, y'(0) = 2.(36)1600a. Solve the initial value problem and plot the solution.b. Determine the coordinates (tM, YM) of the maximum point.c. Change the second initial condition to y'(0) = b > 0 andfind the solution as a function of b.d. Find the coordinates (tM, YM) of the maximum point in termsof b. Describe the dependence of tM and yM on b as b increases.14. Consider the equation ay&#34; + by' + cy= 0. If the roots of thecorresponding characteristic equation are real, show that a solution tothe differential equation either is everywhere zero or else can take onthe value zero at most once.Problems 15 through 17 indicate other ways of finding the secondsolution when the characteristic equation has repeated roots.conIn each oforder to fir15. a. Consider the equation y&#34; + 2ay' + a²y = 0. Show that theroots of the characteristic equation are r₁ r₂ = -a so that onesolution of the equation is eat.b. Use Abel's formula [equation (23) of Section 3.2] to showthat the Wronskian of any two solutions of the given equation isW(t) = y₁(t) y₂(t) - y₁ (1) y₂(t) = c₁e-2at,18. 12y19. Fy20. 1221. xywhere c₁ is a constant.c. Let y₁ (t) =eat and use the result of part b to obtaina differential equation satisfied by a second solution y2(t). Bysolving this equation, show that y2(t) = te-at.22. X23. Tarisescircuand24.equsol

Accepted Answer

y''-6y'+9y=0

5. y"-6y' +9y=0