Test 2 Solutions

Test 2 Solutions - (a-1 √ 3 i Ans 2 e 2 π 3(b 1-i Ans...

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MAT 274 TEST 2 Instructor : Dongrin Kim Name : Score : 1. Find the general solution of each diﬀerential equation. (a) y 00 + 3 y 0 + y = 0 Ans : c 1 e ( - 3+ 5) t/ 2 + c 2 e ( - 3 - 5) t/ 2 (b) y 00 - 8 y 0 + 16 y = 0 Ans : c 1 e 4 t + c 2 te 4 t (c) y 00 + 4 y 0 + 8 y = 0 Ans : c 1 e - 2 t cos2 t + c 2 e - 2 t sin2 t (d) y 000 - 2 y 00 + 2 y 0 - 4 y = 0 Ans : c 1 e 2 t + c 2 e 2 t + c 3 e - 2 t 1

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(e) y (4) + 4 y 00 + 4 y = 0 Ans : c 1 cos 2 t + c 2 sin 2 t + c 3 t cos 2 t + c 4 t sin 2 t (f) y 00 + 2 y 0 - 15 y = 60 Ans : c 1 e 3 t + c 2 e - 5 t - 4 (g) y 00 - 4 y 0 - 5 y = e t Ans : c 1 e - t + c 2 e 5 t - 2 e t (h) y 00 + y = cos t Ans : c 1 cos t + c 2 sin t + t 2 sin t 2
(i) y 00 + 4 y = 2csc2 t, 0 < t < π/ 2 Ans : c 1 cos2 t + c 2 sin2 t + sin2 t 2 ln | sin2 t | - t cos2 t 2. Find the Wronskian of two solutions of the diﬀerential equation (cos x ) y 00 + (sin x ) y 0 - xy = 0 without solving the equation. Ans : c cos x 3. Write in the form of a + bi . (a) e 2 - 3 i Ans : e 2 cos3 - ie 2 sin3 (b) π - 1+2 i Ans : π - 1 cos(2ln π ) + - 1 sin(2ln π ) 3

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4. Write in the form of Re . Give exact answers.
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Unformatted text preview: (a)-1 + √ 3 i Ans : 2 e 2 π/ 3 (b) 1-i Ans : √ 2 e-π/ 4 5. y = t is a solution of t 2 y 00 + 2 ty-2 y = 0. Find the second solution. Ans : t-2 6. Compute the Wronskian W (1 ,t,e-t ,te-t ). Ans : e-2 t 7. Write the particular solution with undetermined coeﬃcients . Do not evalu-ate them. (a) y 00 + y = 1 + e-t Ans : c 1 t 2 + c 2 te-t (b) y 00-2 y + y = te t + t cos t Ans : ( c 1 t 3 + c 2 t 2 ) e t + ( c 3 t + c 4 )cos t + ( c 5 t + c 6 )sin t 4...
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Test 2 Solutions - (a-1 √ 3 i Ans 2 e 2 π 3(b 1-i Ans...

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