MA 30300 SAMPLE MIDTERM EXAM #1 SOLUTIONS Problem 1. Consider the differential equation x3 − x2 y 00 + (x − 1) y 0 + y = 0. a. Find the singular points of the differential equation. b. Classify these singular points as regular or irregular. Explain your reasoning. Solution: (a) We have the coefficients P (x) = x3 − x2 Q(x) = x − 1 R(x) = 1 Q(x) 1 p(x) = P (x) = x2 =⇒ 1 R(x) q(x) = = 2 P (x) x (x − 1) These functions fail to be continuous at x0 = 0 and x0 = 1, so these are the singular points. (b) We follow the discussion of section 5.4 in the text. We compute two pairs of limits: lim (x − x0 ) · p(x) = lim (x − x0 )2 · q(x) = lim (x − x0 ) · p(x) = lim (x − x0 )2 · q(x) = x→x0 lim x · x→0 x0 = 0 : x→x0 x→x0 1 x2 lim x2 · x→0 x2 lim (x − 1) · x→1 x0 = 1 : x→x0 1 (x − 1) ∞ = −1 1 x2 lim (x − 1)2 · x→1 = 1 x2 (x − 1) = 0 = 0 Hence x0 = 0 is an irregular singular point while x0 = 1 is a regular singular point. Problem 2. Consider the differential equation x2 y 00 + 4 x y 0 + x2 + 2 y = 0. a. Find the indicial equation F (r) = 0, as well as the exponents r at the singularity. b. Use Frobenius’ Method to find a solution corresponding to the smaller exponent at the singularity. List the first three terms of your answer via a series in the form " 2 k # ∞ X x r y(x) = x 1 + a2k (r) , x > 0. 2 k=1 1 Solution: We follow the discussion in section 5.6 of the text. Using part (b), we will guess that the solution is a series in the form " 2 k # X ∞ ∞ X x 1 a2k x2k+r . y(x) = xr 1 + = a2k (r) 2 2k k=1 k=0 Then we have the following derivatives: ∞ ∞ X X 1 1 2k+r 2 a x a2k x2k+r+2 x2 + 2 y = + 2k 2k 2k " 4 x y0 = k=0 ∞ X k=0 x2 y 00 = = k=0 1 4 (2 k + r) a2k x2k+r 2k = ∞ X 1 (2 k + r) (2 k + r − 1) a2k x2k+r 2k = k=0 ∞ X # 2 k x xr 2 a0 + 2 a2k + 2 a2k−2 2 k=1 " 2 k # ∞ X x r x 4 r a0 + 4 (2 k + r) a2k 2 k=1 r (r − 1) a0 2 k ∞ xr X x + (2 k + r) (2 k + r − 1) a2k 2 k=1 Adding these together, we find the expression " 2 k # ∞ X x 2 00 0 2 r x y + 4 x y + x + 2 y = x F (r) a0 + F (2k + r) a2k + 2 a2k−2 2 k=1 in terms of F (r) = r (r − 1) + 4 r + 2 = r2 + 3 r + 2 = (r + 1) (r + 2). (a) As we found above, the indicial equation is F (r) = r2 + 3 r + 2 = 0. The exponents at the singularity are just the roots of this equation, so they are r1 = −1 and r2 = −2. (b) We must also have F (2k + r) a2k + 2 a2k−2 = 0 for k ≥ 1. As the smaller exponent is r = −2, we find that 1 F (2k + r) = (2k + r) + 1 (2k + r) + 2 = 2 k − 1 2 k =⇒ a2k = − a2k−2 . (2 k − 1) k The first few terms are −1 (−1)k −1 +1 a0 , a6 = a0 , ··· a2k = a0 . a2 = a0 , a4 = 1·1 1 · 3 · 2! 1 · 3 · 5 · 3! 1 · 3 · · · (2k − 1) · k! Hence the solution is " # 2 2 2 3 2 k x x x 1 x2 1 1 (−1)k y(x) = 2 1 − + − + ··· + + ··· x 2 1 · 3 · 2! 2 1 · 3 · 5 · 3! 2 1 · 3 · · · (2k − 1) · k! 2 Problem 3. Consider the initial value problem y 00 + 6 y 0 + 25 y = 0 such that y(0) = 0 a. Find the Laplace transform L y (s) of the solution y(t). b. Find the solution y(t) to the initial value problem. and y 0 (0) = 4. Solution: (a) Corollary 6.2.2 asserts that L y (s) L y 0 (s) L y 00 (s) = = Y (s) s Y (s) − 2 y(0) = s Y (s) − s y(0) − =⇒ y (0) 0 2 0 = L y 00 + 6 y 0 + 25 y (s) = L y 00 (s) + 6 L y 0 (s) + 25 L y (s) = s2 Y (s) − 4 + 6 s Y (s) + 25 Y (s) Solving for Y (s), we find that 4 . + 6 s + 25 (b) We complete the square in the denominator above to see that b 4 = in terms of Y (s) = (s + 3)2 + 42 (s − a)2 + b2 Using Table 6.2.1, we know that f (t) = eat sin bt =⇒ L f (s) = Y (s) = s2 so we see that the solution is y(t) = e−3t sin 4t. 3 a = −3, b (s − a)2 + b2 b = 4.
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