Linear Homogeneous Differential Equations

SECOND-ORDER DIFFERENTIAL EQUATIONS Exercise Set 14.1 key (14-1) 2nd order Linear Homogeneous Diff egtns 1. Verify that the following are solutions of the differential equation y^ prime prime - y' - 2y = 0 substituting these functions into the equation: (a) e ^ (3x) * and * e ^ (- x) (b) c_{1} * e ^ (2x) + c_{2} * e ^ (- x) ( c_{\mathbb{Z}} constants). 2. Verify that the following are solutions of the differential equation y^ prime prime + 4 * y' + 4y = 0 by substituting these func tions into the equation: (a) epsilon ^ (-2x) and (b) c_{1} * e ^ (-2x) + c_{2}*x * e ^ (-2x) (C. constants). In Exercises 3-16, find the general solution of the differen-tial equation. 3. y^ prime prime + 3 * y' - 4y = 0 5. y^ prime prime - 2 * y' + y = 0 . 7. y^ prime prime + 5y = 0 . 1. (d ^ 2 * y)/(d * t ^ 2) + 4 * d/dt (y) + 4y = 0 4. y^ prime prime + 6 * y' + 5y = 0 . 6. y^ prime prime + 6 * y' + 9y = 0 . 8. y^ prime prime + y = 0 10 (d ^ 3 * y)/(d * x ^ g) + 3 * d/dx (y) = 0 3. (d ^ 2 * y)/(d * x ^ 2) - 4 * d/dx (y) + 13y = 0 (d ^ 2 * y)/(d * x ^ 2) - 6 * d/dx (y) + 25y = 0 5. 8y^ prime prime - 2 * y' - y = 0 . 16. 9y^ prime prime - 6 * y' + y = 0 . In Exercises 17-22, solve the initial-value problem. 7. y ^ x + 2 * y' - 3y = 0 y(0) = 1, y' * (0) = 5 9 . y^ prime prime - 6 * y' + 9y = 0 y(0) = 2, y' * (0) = 1 y^ prime prime - 6 * y' - 7y = 0 ; y(0) = 5, y' * (0) = 3 y^ prime prime + 4 * y' + y = 0 ; r(0) = S, y' * (0) = 4 1. y^ prime prime + 4 * y' + 5y = 0 ; y(0) = - 3, y' * (0) = 0 y^ prime prime - 6 * y' + 13y = 0 ; y(0) = - 1, y' * (0) = 1 In each part find a second-order linear homogeneous differential equation with constant coefficients that has the given functions as solutions. (a) y_{1} = e ^ (5x) (b) y_{1} = e ^ (4x) y_{2} = x * e ^ (4pi) (c) y_{1} = e ^ (- x) * cx 4x, y_{z} = e ^ (- x) * sin 4x Show that if e and are solutions of a second-order linear homogeneous differential equation, then so are cosh x and sinh x Find all values of & for which the differential equation y^ prime prime + k * y' + ky = 0 has a general solution of the given (a) y = c_{1} * e ^ (ax) + c_{2} * e ^ (bx) (b) y = c_{1} * e ^ (ax) + c_{2}*x * e ^ (ax) (c) y = c_{1} * e ^ (ax) * cos bx + c_{2} * e ^ (ax) * sin bx 26. The equation x ^ 2 * (d ^ 2 * y)/(d * x ^ 2) + px * d/dx (y) + qy = 0 0) where p and q are constants, is called Euler'a equi dimensional equation. Show that the substitution transforms this equation into the equation (d ^ 2 * y)/(d * z ^ 2) + (p - 1) * d/dz (y) + qy = 0 27. Use the result in Exercise 26 to find the general solution of (a) x ^ 2 * (d ^ 2 * y)/(d * x ^ 2) + 3x * d/dx (y) + 2y = 0 (b) x ^ 2 * (d ^ 2 * y)/(d * x ^ 3) - x * d/dx (y) - 2y = 0 28. Let be a solution of y^ prime prime + p * y' + qy = 0 . Prove: If p and q are positive constants lim x -> ∞ y(x) = 0 29. The Wronskian of two differentiable functions y 1' and denoted by W(y_{1}, y_{2}) and is defined to be the function W(y_{1}, y_{2}) = y_{1} * y_{2}' - y_{1}' * y_{2} = det [[y_{1}, y_{3}], [y_{1}', y_{2}']] The value of W(y_{1}, y_{2}) at a point is denoted by W(y_{1}, y_{2})(x) or simply by It can be proved that two solutions, y_{1} = y_{1}(x) and y_{2} = y_{2}(x) of Equation (2) are lincarly if if for all . Equivalently, the functions are linearly independent if and only if \mathcal{W}(x) ne0 for at least one value of x. Use this result to prove that the following solutions of Equation (4) are linearly independent: (4) y_{1} = e ^ (m_{0}*x) y_{2} = e ^ (m_{1}*x) (b) y_{1} = e ^ (mx) y_{2} = x * e ^ (mx) 30. Prove: If the auxiliary equation of y^ prime prime + p * y' + qy = 0 has complex roots and then the general solution of this differential equation is y(x) = e ^ (ax) * (c_{1} * cos bx + c_{2} * sin bx) [Hint: By substitution, verify that y_{1} = epsilon ^ (ax) * cos bx and bx are solutions of the equation. Exercise 29 to prove that and are linearly independent.] 31. Suppose that the auxiliary equation of the equation y^ prime prime + p * y' + qy = 0 has distinct real roots and m. (a) Show that the function is a solution of the differential equation. g_{mu}(x) = (e ^ (mu*x) - e ^ (mx))/(mu - m) (d ^ 2 * y)/(d * t ^ 2) - 10 * d/dt (y) + 25y = 0 (d ^ 2 * y)/(d * x ^ 0) - d/dx (y) = 0 ( m_{1} ne m 2 ) Chpt. 12 Differential Equations (14-1) Second-order Linear Homogeneous Differentialegins. P.662 \# (4 - 18) even & al y^ prime prime + 6 * y' + 5y = 0 m ^ 22 + 6m + 5 = 0 (m + 5)(m + 1) = 0 y = c_{1} * e ^ (-5R) + c_{2} * e ^ (- x) 0y y^ prime prime + 6 * y' + 9y = 0 (m ^ 2 + (mr) + 9) = 0 (m+2)( m + z = 0 m = - 2 y = c_{1} * e ^ (-3x) + c_{2}*x * e ^ - 8 y^ prime prime + y = 0 m ^ 2 + 11 = 0 m ^ 2 = - i m = plus/minus L i =\ 5 alpha = 0 b=1 y = e ^ (0x) * (c_{1} * cos x + C_{2} * sin x) y = (c_{1} * cos x + C_{2} * sin x) • (d ^ 2 * y)/(d * x ^ 2) + 9/(dy) = 0 y^ prime prime + 3 * y' = 0 m ^ 2 + 3m = 0 m(m + 2) = 0 m m = - 2 y = c_{1} * e ^ (0x) + c_{2} * e ^ (-3x) y = c_{1} + c_{2} * e ^ (-3x) (12) (d ^ 2 * y)/(d * t ^ 2) - 10 * d/dt (y) + 25y = 0 y^ prime prime - 10y ^ 1 + 25y = 0 m ^ 2 - 10m + 25 = 0 (m - 5)(m - 5) = 0 m = 5 14. (d deg * y)/(d * x ^ 2) - 6 * d/dx (y) + 25y = 0 y^ prime prime - 6 * y' + 25y = 0 r * m ^ 2 - 6m + 25 = 0 m_{1} = 6 + sqrt(36 - 100) 2 m_{4} = (6 + 8L)/2 = 3 + 4k m_{0} = (6 - sqrt(36 - 10))/2 m_{2} = 3 = 4i Et st y = c_{1}*e + c_{0}*ze a = 3 b = 4 y = e ^ (3x) * (c_{1} * cos 4x + c_{2}*sin * 4x) 16. 9y^ prime prime - 6 * y' + y = 0 (3m - 1)(3m - 1) = 0 9m ^ 2 - 6m + 1 = 0 18. y^ prime prime - 6 * y' - 7y = 0 y(o) = 5 y' * (0) = 3 (m - 7)(m + 1) = 0 m = 7 m = - 1 y = c_{1} * e ^ (7x) + c_{2} * e ^ (- x) 5 = C_{1} + C_{2} Ce 5 = c_{1}*e y = c_{1}*e C_{D}*e m = 1/2 y(0) = - 3 y^ prime prime + 4 * y' + 5y = 0 overline y' * (0) = 0 m ^ 2 + 4m + 5 = 0 05 5 = C_{1} + C_{2} m_{2} = - 2 - k G = - 2 y = 1 y = e ^ (-2x) * (c_{1} * cos x + c_{0} * sin x) (C 1 ^ cos(0) + C_{2} <= 15/0 ) - 3 = e ^ (-2(0)) m_{1} = (- 4 + 2L)/2 = - 2 + L m_{1} = (- 4 + sqrt(16 - 20))/2 * m_{2} = (- 4 - sqrt(16 - 20))/2 3 = 7C_{12} - C_{2} y^ prime = 7c_{1} * e ^ (7x) -c 2 c^ - 3 = 7c_{1} * e ^ 710 - c_{3} * e ^ (- nu) 8 = 8c_{1} C_{1} = 1 y = e ^ (7x) + 4e ^ (- x ^ 2) - 3 = C_{1} y' = - 2e ^ (- - x) * (c_{1} * cos x + c_{2}*sinx) + e ^ (-2x) * (- c_{1} * sin x + c_{2} * cos x) 0 = - 2(- 3) + 1(C_{2}) O^ prime = - 2e ^ (-2(0)) * (- 3cos(0) + c_{2} * sin(0)) +e^ -2(0) (+35+c 2 cos(0) 0 =6+C0 6 = C_{2} -2x y = - e * (3osx + 6sin x) 5 = 1 + C_{2} C_{2} = 4 3 = 7C_{1} - C_{2} y = e ^ (-2x) * (- 3cos x - 6sin x) * gr