SOTUIONS Iipart (a).]ane for whichplution whose32. (a) Show that a solution from the family in part (a) of Problem 31that satisfies y' = y², y(1) = 1, is y = 1/(2 – x).(b) Then show that a solution from the family in part (a)of Problem 31 that satisfies y' = y², y(3) = –1, isy = 1/(2 – x).(c) Are the solutions in parts (a) and (b) the same?33. (a) Verify that 3x² – y² = c is a one-parameter family ofsolutions of the differential equation y dy/dx = 3x.(b) By hand, sketch the graph of the implicit solution3x – y = 3. Find all explicit solutions y = 4(x) of thein part (a) defined by this relation. Give the interval Iof definition of each explicit solution.uarantees thatnique solution(c) The point (-2, 3) is on the graph of 3x2 – y² = 3,but which of the explicit solutions in part (b) satisfiesy(-2) = 3?solutions of34. (a) Use the family of solutions in part (a) of Problem 33 toch memberroblemfind an implicit solution of the initial-value problemydy/dx = 3x, y(2) = -4. Then, by hand, sketch thegraph of the explicit solution of this problem and give itshe xy-planeuld have ainterval I of definition.= 3x that pass(b) Are there any explicit solutions of y dy/dxthrough the origin?In Problems 35–38 the graph of a member of a family of solutionsof a second-order differential equation dy/dx2 = f(x, y, y') is given.Match the solution curve with at least one pair of the following initialconditions.ether this(a) y(1) = 1, y'(1) = -2problem in(b) y(-1) = 0, y'(-1) = -4family ofy?.(c) y(1) = 1, y'(1) = 2(d) y(0) = -1, y'(0) = 2tinuous ev-(e) y(0) = -1, y'(0) = 0e taken toons in part· initial-valueo = 0 is inis not defined(f) y(0) = -4, y'(0) = -235.pr the solutionfamily ofхous every-aken to beamily inution fromFIGURE 1.2.7 Graph for Problem 35DetermineRights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202

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