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# Class Note for MATH 3321 at UH

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Date Created: 02/06/15

13 Initial Conditions InitialValue Problems As we noted in the preceding section we can obtain a particular solution of an nth order differential equation simply by assigning speci c values to the n constants in the general solution However in typical applications of differential equations you will be asked to nd a solution of a given equation that satis es certain preassigned conditions Example 1 Find a solution of 2 y 31 7 21 that passes through the point l 3 SOLUTION In this case we can nd the general solution by integrating y31272z dzzg712Ci The general solution is y 13 7 12 C To nd a solution that passes through the point 2 6 we set I 2 and y 6 in the general solution and solve for C 623722C874C whichimplies 02 3 Thus y z 7 12 2 is a solution of the differential equation that satis es the given condition In fact it is the only solution that satis es the condition since the general solution represented all solutions of the equation and the constant C was uniquely determined I Example 2 Find a solution of 12y 7 21y 2y 413 which passes through the point 1 4 with slope 2 SOLUTION As shown in Example 4 in the preceding section the general solution of the differential equation is y C112 C21 213 Setting 1 l and y 4 in the general solution yields the equation C1 C2 2 4 which implies C1 C2 2 The second condition slope 2 at z l is a condition on y We want y l 2 We calculate y y 2C11 C2 612 and then set I l and y 2 This yields the equation 201 C2 6 2 which implies 2C1 02 4 Now we solve the two equations simultaneously 11 C1 C2 2 2C1 C2 74 We get C1 76 C2 8 A solution of the differential equation satisfying the two conditions is 7612 81 213 It will follow from our work in Chapter 3 that this is the only solution of the differential equation that satis es the given conditions I INITIAL CONDITIONS Conditions such as those imposed on the solutions in Examples 1 and 2 are called initial conditions This term originated with applications where processes are usually observed over time starting with some initial state at time t 0 Example 3 The position yt of a weight suspended on a spring and oscillating up and down is governed by the differential equation y 9y 0 a Show that the general solution of the differential equation is yt C1 sin 3t C2 cos 3t b Find a solution that satis es the initial conditions y0 l y 0 72 SOLUTION a y C1 sin 3t C2 cos 3t y 3C1 cos 3t 7 SC sin 3t y 79C1 sin 3t 7 9C2 cos 3t Substituting into the differential equation we get y 9y 79C1 sin 3t 7 9C2 cos 3t 9 C1 sin 3t C2 cos 3t 0 Thus yt C1 sin 3t C2 cos St is the general solution b Applying the initial conditions we obtain the pair of equations y0 l C1 sin 0 C2 cos 0 C2 which implies C2 l yO 72 3C1 cos 0 7 SC sin 0 which implies C1 7 who A solution which satis es the initial conditions is yt 7 sin 3t cos St I Any nth order differential equation with independent variable I and unknown function y can be written in the form F y39vy wym lhymd 7 0 1 12 by moving all the nonzero terms to the lefthand side Since we are talking about an nth order equation gm must appear explicitly in the expression F Each of the other arguments may or may not appear explicitly For example the thirdorder differential equation 12y 7 21y few written in the form of equation 1 is 12y 7 21y 7 few 0 2y and Fzyy y ym z 7 21y 7 erTy Note that y does not appear explicitly in the equation However it is there implicitly For example y y y y nth ORDER INITIALVALUE PROBLEM An nth order initialvalue problem consists of an n th order differential equation FIyy7y 7y WW 0 together with n initial conditions of the form MC 1607 40 k1 y 6 k2 WAVE 767171 where c and kg 161 16771 are given numbers It is important to understand that to be an nth order initialvalue problem there must be n conditions same n of exactly the form indicated in the de nition For example the problem 0 Find a solution of the differential equation y 9y 0 satisfying the conditions y0 0 y7r 0 is not an initialvalueproblem the two conditions are not of the form in the de nition Similarly the problem 0 Find a solution of the differential equation y 7 3y 7 y 0 satisfying the conditions y0 l y 0 2 is not an initialvalue problem a third order equation requires three conditions yc k0 y c k1 y c k2 EXISTENCE AND UNIQUENESS The fundamental questions in any course on differential equations are 1 Does a given initialvalue problem have a solution That is do solutions to the problem exist 2 If a solution does exist is it unique That is is there exactly one solution to the problem or is there more than one solution 13 The initialvalue problems in Examples 17 2 and 3 each had a unique solution values for the arbitrary constants in the general solution were uniquely determined Example 4 The function y 12 is a solution of the differential equation y Z and y0 0 Thus the initialvalue problem y 2W y0 0 has a solution However7 y E 0 also satis es the differential equation and y0 0i Thus7 the initialvalue problem does not have a unique solution In fact7 for any positive number a the function 07 I g a yaw 17a7 zgta is a solution of the initialvalue problemi I Y Example 5 The oneparameter family of functions y CI is the general solution of There is no solution that satis es y0 l the initialvalue problem 9 y 7 W 1 does not have a solution I The questions of existence and uniqueness of solutions Will be addressed in the speci c cases of interest to us A general treatment of existence and uniqueness of solutions of initialvalue problems is beyond the scope of this course 14 Exercises 13 1 a b b b b C b b Show that each member of the oneparameter family of functions y 065x is a solution of the differential equation y 7 5y 0 Find a solution of the initial value problem y 7 5y 07 y0 2 Show that each member of the twoparameter family of functions y 01621 02671 is a solution of the differential equation y 7 y 7 2y 0 Find a solution of the initial value problem y 7 y 7 2y 0 y0 2 y 0 1 Show that each member of the oneparameter family of functions 7 1 y 7 Ce l is a solution of the differential equation y y y2l Find a solution of the initial value problem y y y2 yl 71 Show that each member of the threeparameter family of functions y 0212 C11 C0 is a solution of the differential equation y 0 Find a solution of the initial value problem y 0 yl l7 y l 4 y l 2 Find a solution of the initial value problem y 0 y2 y 2 y 2 0 Show that each member of the twoparameter family of functions y C1 sin 31 C2 cos 31 is a solution of the differential equation y 9y 0 Find a solution of the initial value problem y 9y 0 y7r2 y 7r2 1 Show that each member of the twoparameter family of functions y C112 C212 ln 1 is a solution of the differential equation 12y 7 3Iy 4y 0 Find a solution of the initial value problem 12y 7 3Iy 4y 0 yl 07 y l 1 Is there a member of the twoparameter family Which satis es the initial condition y0 y 0 0 15 d Is there a member of the twoparameter family Which satis es the initial condition y0 07 y 0 1 If not7 Why not 7 a Show that each member of the twoparameter family of functions y C11 C2zl2 is a solution of the differential equation 212g 7 my y 0 b Find a solution of the initial value problem 212g 7 zy y 0 y4 l7 y 4 72 c Is there a member of the twoparameter family Which satis es the initial condition y0 l7 yO 2 If not7 Why not 8 Each member of the twoparameter family of functions y Clsin zC2cos z is a solution of the differential equation y y 0 a Determine Whether there are one or more members of this family that satisfy the condi tions ylt0gt o W 0 ow a e zero unc lon7 y E 1s e on y mem er 0 e am1 y a sa 1s es e b Sh th tth f t39 0 39 th 1 b fth f 391 th t t39 th conditions MO 07 MTV2 0 9 Given the differential equation y 2 7 zy y 0 a Show that the family of straight lines y CI 7 C2 is the general solution of the equation b Show that y i 12 is a solution of the equation Note that this function is not included in the general solution of the equation it is a singular solution of the equation 10 Given the differential equation 1y 2 7 ny 4x 0i 12C2 a Show that the oneparameter family y T is the general solution of the equation b Show that each of y 21 and y 721 is a solution of the equation Note that these functions are not included in the general solution of the equation they are singular solutions of the equation Find the differential equation of the given family 11 y 013 1i 12 y C12 3 13 y 0621 6 21 14 y Ce sin 1 15 y C1 C 16 y Clem Cge hi 16 17 18 19 20 21 22 23 24 25 y C1 C21e2 y C11 C21 y C11 C21 1i y C1 cos 31 C2 sin 31 y C1 sin 31 C2 y C162 cos 31 C2 y C1 C21 C31 y C11 C212 C313i Verify that the function ya1 is a solution of the initialvalue problem y Z y0 01 17

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