Part I Problems and Solutions

Part I Problems and Solutions Find the general solution to the given DE and also the specific solution satisfying the given initial conditions (if any). Problem 1: y�� − 3y� + 2y = 0 Solution: Characteristic equation p(r ) = r2 − 3r + 2 = 0 or (r − 1)(r − 2) = 0 with roots r = 1, 2. Thus we have the general solution y = c1 e x + c2 e2x Problem 2: y�� + 2y� − 3y = 0, y(0) = 1, y� (0) = −1. Solution: Characteristic equation r2 + 2r − 3 = 0, or (r + 3)(r − 1) = 0, so y = c1 e x + c2 e−3x . Put in initial conditions: y (0) = 1 ⇒ c1 + c2 = 1 � y (0) = −1 ⇒ c1 − 3c2 = −1 Solve for c1 , c2 , and we get y= 1 x 1 −3x e + e 2 2 In the next three problems, find a DE of the form ay�� + by� + cy = 0 which has the given family of solutions y = c1 y1 + c2 y2 , with c1 , c2 constant. Problem 3: y = c1 + c2 e−5x Solution: y = c1 er1 x + c2 er2 x with r1 , r2 roots of p(r ) = ar2 + br + c → P(r ) = (r − r1 )(r − r2 ). In this problem, r1 = 0, r2 = −5, so p(r ) = (r − 0)(r + 5) = r2 + 5r. Thus, a = 1, b = 5, c = 0, so the DE y�� + 5y� = 0 has these solutions. Problem 4: y = c1 e5x + c2 e−5x Solution: r1 = 5, r2 = −5 so p(r ) = (r − 5)(r + 5) = r2 − 25 = ar2 + br + c so a = 1, b = 0, c = −25, and so the DE y�� − 25y = 0 has these solutions. Problem 5: y = c1 + c2 x OCW 18.03SC Solution: y� = c2 and y�� = 0, so we can take a = 1, b = c = 0. Thus the DE y�� = 0 has these solutions. In the next four problems, find the general solution of the given DE. Problem 6: y�� − 4y = 0 Solution: Characteristic equation p(r ) = r2 − 4 = 0 so roots are ±2, and so y = c1 e2x + c2 e−2x is the general solution. Problem 7: 2y�� − 3y� = 0 Solution: Characteristic equation p(r ) = 2r2 − 3r = 0 has roots 0, 32 , and so the general solution is 3 y = c1 + c2 e 2 x Problem 8: 4y�� − 12y� + 9y = 0 Solution: Characteristic equation p(r ) = 4r2 − 12r + 9 = (2r − 3)2 = 0 so we have one repeated root r = 32 . Thus the solutions are 3 3 y = c1 e 2 x + c2 xe 2 x Problem 9: y(4) − 8y�� + 16y = 0 Solution: Characteristic equation p(r ) = r4 − 8r2 + 16 = (r2 − 4)2 = (r − 2)2 (r + 2)2 = 0. This has double roots at r = ±2, so the solutions are: y = c1 e2x + c2 xe2x + c3 e−2x + c4 xe−2x Find the general solution to the general DE and also the one satisfying the initial conditions (if any are given). Problem 10: y�� + 2y� + 2y = 0 Solution: Char. eqn. r2 + 2r + 2 = 0 By quadratic formula, r = −1 ± i, so the general solution is y = e− x (c1 cos x + c2 sin x ) OCW 18.03SC (using as y1 , y2 the real and imaginary parts of the characteristic solution y = e(−1+i)x = e− x (cos x + i sin x )) Problem 11: y�� − 2y� + 5y = 0; y(0) = 1, y� (0) = −1 Solution: Characteristic equation r2 − 2r + 5 = 0. By quadratic formula, r = 1 ± 2i. General solution is thus y = e x (c1 cos 2x + c2 sin 2x ) Putting in initial conditions (you’ll need to find y� first!): y (0) = 1 ⇒ c1 = 1 y� (0) = 1 ⇒ c1 + 2c2 = −1 → c2 = −1 so y = e x (cos 2x − sin 2x ) Problem 12: y�� − 4y� + 4y = 0; y(0) = 1, y� (0) = 1 Solution: Characteristic equation r2 − 4r + 4 = 0 or (r − 2)2 = 0; r = 2 double root. So y = e2x (c1 x + c2 ) is the general solution. Put in initial conditions: y (0) = 1 ⇒ c2 = 1 y� (0) = 1 ⇒ 2c2 + c1 = 1 → c1 = −1 So the solution is y = (1 − x )e2x . Problem 13: Find the general solution to the DE y�� + 6y� + 9y = 0 Solution: Characteristic equation r2 + 6r + 9 = (r + 3)2 = 0 has a double root r = −3 so the general solution is y = c1 e−3x + c2 xe−3x . In the next two problems, solve the given initial-value problem. Problem 14: y�� − 4y� + 3y = 0, y(0) = 7, y� (0) = 11. OCW 18.03SC Solution: Characteristic equation r2 − 4r + 3 = 0 → (r − 3)(r − 1) = 0 with roots r = 1, 3 so the general solution to this DE is y = c1 e x + c2 e3x IC’s: y (0) = c1 + c2 = 7 c1 = 5 ⇒ � y (0) = c1 + 3c2 = 11 Problem 15: y�� − 6y� + 25y = 0, ⇒ y = 5e x + 2e3x c2 = 2 y(0) = 3, y� (0) = 1 Solution: Characteristic equation r2 − 6r + 25 = 0 has roots r = 1 2 (6 ± 8i ) = 3 ± 4i. 1 2 � 6± √ � 36 − 100 = Real solutions are y = e3x (c1 cos 4x + c2 sin 4x ). IC’s: y(0) = c1 = 3, y� (0) = 3c1 + 4c2 = 1 gives c1 = 3, c2 = −2, so the solution is y = e3x (3 cos 4x − 2 sin 4x ) Problem 16: For the equation y�� + 2y� + cy = 0, c constant, a) Tell which values of c correspond to each of the three cases: two real roots, repeated real root, and complex roots. b) For the case of two real roots, tell for which values of c both roots are negative, both roots are positive, or the roots have different signs. c) Summarize the above information by drawing a c-axis, and marking the intervals on it corresponding to the different possibilities for the roots of the characteristic equation. d) Finally, use this information to mark the interval on the c-axis for which the correspond­ ing ODE is stable. (The stability criterion using roots is what you will need.) Solution: y�� + 2y� + cy = 0 has characteristic equation r2 + 2r + c = 0, with roots −1 ± √ 1 − c. Below 0, there is one negative and one positive root. At 0, there is one root. Between 0 and 1, there are two real negative roots. At 1, there is one negative root. Greater than 1, there are two complex roots with negative real part. The region below 0 is unstable; the rest of the axis is stable. OCW 18.03SC