9 Returning to the equation ay" + by' + cy = 0 with arbitrary coefficients, recallthat when r₁ r2, its general solution (18) is the sum of two exponential functions.Therefore, the solution has a relatively simple geometrical behavior: as t increases, themagnitude of the solution either tends to zero (when both exponents are negative) or elseexhibits unbounded growth (when at least one exponent is positive). These two casesare illustrated by the solutions of Examples 3 and 4, which are shown in Figures 3.1.1and 3.1.2, respectively. Note that whether a growing solution approaches +∞or - as1 → ∞ is determined by the sign of the coefficient of the exponential for the larger rootof the characteristic equation. (See Problem 21.) There is also a third case that occursnegative.less often: the solution approaches a constant when one exponent is zero and the other isGIn Sections 3.3 and 3.4, respectively, we return to the problem of solving the equationay" +by' + cy=0 when the roots of the characteristic equation either are complex conjugatesor are real and equal. In the meantime, in Section 3.2, we provide a systematic discussionequations.of the mathematical structure of the solutions of all second-order linear homogeneousProblemsIn each of Problems 1 through 6, find the general solution of the givendifferential equation.1. y" + 2y' - 3y = 01)+ (1Enia)+(182.y" + 3y' + 2y = 01)+183.6y" - y'- y = 04.y" + 5y' = 05.4y" - 9y = 06. y" - 2y' - 2y = 0In each of Problems 7 through 12, find the solution of the given 19.initial value problem. Sketch the graph of the solution and describeits behavior as t increases.G 7.8.badria10 mie 0-=0q+³0=3py"+y' - 2y = 0,y" +4y' + 3y = 0,y" + 3y' = 0, y(0)2y"+y'- 4y = 0,y" +8y' - 9y = 0,y(0) = 1, y'(0) = 1y(0) = 2, y'(0) = -1= -2, y'(0) = 3y(0) = 0, y'(0) = 1y(1) = 1, y'(1) = 0y(-2) = 1, y'(-2) = -1G 9.G 10.G 11.G 12. 4y" - y = 0,13. Find a differential equation whose general solution isy=c₁e²¹ +c₂e-³1.101 110G 14. Find the solution of the initial value problem5y" - y = 0, y(0)=, y'(0) = -34y(0) = 2, y'(0)=SH12Plot the solution for 0 ≤ t ≤ 2 and determine its minimum value.15. Find the solution of the initial value problem2y" - 3y + y = 0,Then determine the maximum value of the solution and also find thepoint where the solution is zero.3.1 Homogeneous Differential Equations with Constant Coefficients 109Inpassyllups16. Solve the initial value problem y" - y' - 2y = 0, y(0) = a,y'(0) = 2. Then find a so that the solution approaches zero ast→∞.-17. y" - (2a - 1) y' + a(a − 1) y = 018. y" +(3-a) y' - 2(a - 1) y = 0mojadi maIn each of Problems 17 and 18, determine the values of a, if any, forwhich all solutions tend to zero as to; also determine the valuesof a, if any, for which all (nonzero) solutions become unbounded ast →∞.Consider the initial value problem (see Example 5)y" + 5y' + 6y = 0, y(0) = 2, y'(0) = 3,where 3 > 0. olsua. Solve the initial value problem.b. Determine the coordinates tm and ym of the maximum pointof the solution as functions of 3.c. Determine the smallest value of 3 for which ym ≥ 4.d. Determine the behavior of tm and ym as 3 →∞0.20. Consider the equation ay" +by' + cy=d, where a, b, c, and dare constants.bs not odjoda. Find all equilibrium, or constant, solutions of this differentialequation.Tb. Let ye denote an equilibrium solution, and let y = y - ye.Thus Y is the deviation of a solution y from an equilibriumsolution. Find the differential equation satisfied by Y.21. Consider the equation ay" + by' + cy = 0, where a, b, and care constants with a > 0. Find conditions on a, b, and c such that theroots of the characteristic equation are:a. real, different, and negative.b. real with opposite signs.c. real, different, and positive.In each case, determine the behavior of the solution as t→∞.

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In each of Problems 7 through 12, find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as t increases. G 7. y"+y'- 2y = 0, G 8. y" +4y' + 3y = 0, y" + 3y = 0, y(0) = -2, y'(0) = 3 (60) G G 9. 10 n y(0) = 1, y'(0) = 15 y(0) = 2, y'(0) = -1