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The value of y 1 ′ = 4 y 2 − 8 cos 4 t , y 2 ′ = − 3 y 1 − 9 sin 4 t with initial condition y 1 ( 0 ) = 0 , y 2 ( 0 ) = 3 . The solution of differential equation is y 1 = sin 4 t , y 2 = 3 cos 4 t _ . Formula used: Crammer rule: The solution of the equation a x + b y = e , c x + d y = f is x = | e b f d | | a b c d | , y = | a e c f | | a b c d | . Calculation: Taking the Laplace transform of the given differential equation y 1 ′ = 4 y 2 − 8 cos 4 t . L [ y 1 ′ ] = L [ 4 y 2 − 8 cos 4 t ] s L ( y 1 ) − y 1 ( 0 ) = 4 y 2 ( s ) − 8 s s 2 + 16 s L ( y 1 ) = 4 y 2 ( s ) − 8 s s 2 + 16 s L ( y 1 ) − 4 y 2 ( s ) = − 8 s 2 + 16 Similarly, taking the Laplace transform of the given differential equation y 2 ′ = − 3 y 1 − 9 sin 4 t . L [ y 2 ′ ] = L [ − 3 y 1 − 9 sin 4 t ] s y 2 ( s ) − y 2 ( 0 ) = − 3 L ( y 1 ) − 9 × 4 s 2 + 16 s y 2 ( s ) − 3 = − 3 L ( y 1 ) − 36 s 2 + 16 3 L ( y 1 ) + s y 2 ( s ) = 3 − 36 s 2 + 16 By the use of Crammer rule, the value of L ( y 1 ) is calculated as: L ( y 1 ) = | − 8 s s 2 + 16 − 4 3 − 36 s 2 + 16 s | | s − 4 3 s | = − 8 s 2 s 2 + 16 + 4 ( 3 − 36 s 2 + 16 ) s 2 + 12 = − 8 s 2 s 2 + 16 + ( 12 − 144 s 2 + 16 ) s 2 + 12 = − 8 s 2 s 2 + 16 + ( 12 − 144 s 2 + 16 ) s 2 + 12 On the further simplification, the following is used. L ( y 1 ) = − 8 s 2 s 2 + 16 + ( 12 ( s 2 + 16 ) − 144 s 2 + 16 ) s 2 + 12 = ( − 8 s 2 + 12 ( s 2 + 16 ) − 144 s 2 + 16 ) s 2 + 12 = ( − 8 s 2 + 12 ( s 2 + 16 ) − 144 ) ( s 2 + 12 ) ( s 2 + 16 ) = ( 4 s 2 + 192 − 144 ) ( s 2 + 12 ) ( s 2 + 16 ) On the further simplification, the following is used. L ( y 1 ) = ( 4 s 2 + 48 ) ( s 2 + 12 ) ( s 2 + 16 ) = 4 ( s 2 + 12 ) ( s 2 + 12 ) ( s 2 + 16 ) = 4 ( s 2 + 16 ) Now the value of y 1 is calculated as: y 1 = L − 1 [ 4 ( s 2 + 16 ) ] = sin 4 t Substitute y 1 = sin 4 t in the equation 4 y 2 = y 1 + 8 cos 4 t . 4 y 2 = y 1 ′ + 8 cos 4 t 4 y 2 = ( 4 cos 4 t ) + 8 cos 4 t 4 y 2 = 12 cos 4 t y 2 = 3 cos 4 t Therefore, the solution of differential equation is y 1 = sin 4 t , y 2 = 3 cos 4 t _ .

The value of y 1 ′ = 4 y 2 − 8 cos 4 t , y 2 ′ = − 3 y 1 − 9 sin 4 t with initial condition y 1 ( 0 ) = 0 , y 2 ( 0 ) = 3 . The solution of differential equation is y 1 = sin 4 t , y 2 = 3 cos 4 t _ . Formula used: Crammer rule: The solution of the equation a x + b y = e , c x + d y = f is x = | e b f d | | a b c d | , y = | a e c f | | a b c d | . Calculation: Taking the Laplace transform of the given differential equation y 1 ′ = 4 y 2 − 8 cos 4 t . L [ y 1 ′ ] = L [ 4 y 2 − 8 cos 4 t ] s L ( y 1 ) − y 1 ( 0 ) = 4 y 2 ( s ) − 8 s s 2 + 16 s L ( y 1 ) = 4 y 2 ( s ) − 8 s s 2 + 16 s L ( y 1 ) − 4 y 2 ( s ) = − 8 s 2 + 16 Similarly, taking the Laplace transform of the given differential equation y 2 ′ = − 3 y 1 − 9 sin 4 t . L [ y 2 ′ ] = L [ − 3 y 1 − 9 sin 4 t ] s y 2 ( s ) − y 2 ( 0 ) = − 3 L ( y 1 ) − 9 × 4 s 2 + 16 s y 2 ( s ) − 3 = − 3 L ( y 1 ) − 36 s 2 + 16 3 L ( y 1 ) + s y 2 ( s ) = 3 − 36 s 2 + 16 By the use of Crammer rule, the value of L ( y 1 ) is calculated as: L ( y 1 ) = | − 8 s s 2 + 16 − 4 3 − 36 s 2 + 16 s | | s − 4 3 s | = − 8 s 2 s 2 + 16 + 4 ( 3 − 36 s 2 + 16 ) s 2 + 12 = − 8 s 2 s 2 + 16 + ( 12 − 144 s 2 + 16 ) s 2 + 12 = − 8 s 2 s 2 + 16 + ( 12 − 144 s 2 + 16 ) s 2 + 12 On the further simplification, the following is used. L ( y 1 ) = − 8 s 2 s 2 + 16 + ( 12 ( s 2 + 16 ) − 144 s 2 + 16 ) s 2 + 12 = ( − 8 s 2 + 12 ( s 2 + 16 ) − 144 s 2 + 16 ) s 2 + 12 = ( − 8 s 2 + 12 ( s 2 + 16 ) − 144 ) ( s 2 + 12 ) ( s 2 + 16 ) = ( 4 s 2 + 192 − 144 ) ( s 2 + 12 ) ( s 2 + 16 ) On the further simplification, the following is used. L ( y 1 ) = ( 4 s 2 + 48 ) ( s 2 + 12 ) ( s 2 + 16 ) = 4 ( s 2 + 12 ) ( s 2 + 12 ) ( s 2 + 16 ) = 4 ( s 2 + 16 ) Now the value of y 1 is calculated as: y 1 = L − 1 [ 4 ( s 2 + 16 ) ] = sin 4 t Substitute y 1 = sin 4 t in the equation 4 y 2 = y 1 + 8 cos 4 t . 4 y 2 = y 1 ′ + 8 cos 4 t 4 y 2 = ( 4 cos 4 t ) + 8 cos 4 t 4 y 2 = 12 cos 4 t y 2 = 3 cos 4 t Therefore, the solution of differential equation is y 1 = sin 4 t , y 2 = 3 cos 4 t _ .

Advanced Engineering Mathematics

Advanced Engineering Mathematics

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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10 5 Obtain by multiplying matrices the composite coordinate transformation of two transformations, first x' = (x + y√√2+2)/2 y' = z' (x√√2-2√2)/2 z = (-x+y√√2-2)/2 followed by x" = (x'√√2+z'√√2)/2 y" = (-x'y'√√2+2')/2 z" = (x'y'√√2-2')/2.

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Chapter 6 Solutions

Advanced Engineering Mathematics

Ch. 6.1 - Prob. 1P Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....

Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Find the transform. Show the details of your work....Ch. 6.1 - Table 6.1. Convert this table to a table for...Ch. 6.1 - Using in Prob. 10, find , where f1(t) = 0 if t ≦...Ch. 6.1 - Table 6.1. Derive formula 6 from formulas 9 and...Ch. 6.1 - Nonexistence. Show that does not satisfy a...Ch. 6.1 - Nonexistence. Give simple examples of functions...Ch. 6.1 - Existence. Show that . [Use (30) in App. 3.1.]...Ch. 6.1 - Change of scale. If and c is any positive...Ch. 6.1 - Inverse transform. Prove that is linear. Hint:...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - Given F(s) = ℒ(f), find f(t). a, b, L, n are...Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.1 - In Probs. 33–36 find the transform. In Probs....Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the IVPs by the Laplace transform. If...Ch. 6.2 - Solve the shifted data IVPs by the Laplace...Ch. 6.2 - Solve the shifted data IVPs by the Laplace...Ch. 6.2 - Solve the shifted data IVPs by the Laplace...Ch. 6.2 - Solve the shifted data IVPs by the Laplace...Ch. 6.2 - Using (1) or (2), find if f(t) equals: t cos 4t Ch. 6.2 - Using (1) or (2), find if f(t) equals: te−at Ch. 6.2 - Using (1) or (2), find if f(t) equals: cos2 2t Ch. 6.2 - Using (1) or (2), find if f(t) equals: sin2 ωt Ch. 6.2 - Using (1) or (2), find if f(t) equals: sin4 t....Ch. 6.2 - Using (1) or (2), find if f(t) equals: cosh2 t Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.2 - INVERSE TRANSFORMS BY INTEGRATION Using Theorem 3,...Ch. 6.3 - Report on Shifting Theorems. Explain and compare...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Sketch or graph the given function, which is...Ch. 6.3 - Find and sketch or graph f(t) if equals e−3s/(s −...Ch. 6.3 - Prob. 13P Ch. 6.3 - Prob. 14P Ch. 6.3 - Find and sketch or graph f(t) if equals e−3s/s4 Ch. 6.3 - Prob. 16P Ch. 6.3 - Prob. 17P Ch. 6.3 - Using the Laplace transform and showing the...Ch. 6.3 - Using the Laplace transform and showing the...Ch. 6.3 - Prob. 20P Ch. 6.3 - Using the Laplace transform and showing the...Ch. 6.3 - Using the Laplace transform and showing the...Ch. 6.3 - Prob. 23P Ch. 6.3 - Using the Laplace transform and showing the...Ch. 6.3 - Prob. 25P Ch. 6.3 - Prob. 26P Ch. 6.3 - Prob. 27P Ch. 6.3 - Prob. 28P Ch. 6.3 - Prob. 29P Ch. 6.3 - Prob. 30P Ch. 6.3 - Prob. 31P Ch. 6.3 - Prob. 32P Ch. 6.3 - Prob. 33P Ch. 6.3 - Prob. 34P Ch. 6.3 - Prob. 35P Ch. 6.3 - Prob. 36P Ch. 6.3 - Prob. 37P Ch. 6.3 - Prob. 38P Ch. 6.3 - Prob. 39P Ch. 6.3 - Prob. 40P Ch. 6.4 - Prob. 3P Ch. 6.4 - Prob. 4P Ch. 6.4 - Prob. 5P Ch. 6.4 - Prob. 6P Ch. 6.4 - Prob. 7P Ch. 6.4 - Prob. 8P Ch. 6.4 - Prob. 9P Ch. 6.4 - Prob. 11P Ch. 6.4 - Prob. 15P Ch. 6.5 - CONVOLUTIONS BY INTEGRATION Find: Ch. 6.5 - CONVOLUTIONS BY INTEGRATION Find: 2. Ch. 6.5 - CONVOLUTIONS BY INTEGRATION Find: 3. Ch. 6.5 - CONVOLUTIONS BY INTEGRATION Find: 4. Ch. 6.5 - Prob. 5P Ch. 6.5 - Prob. 6P Ch. 6.5 - Prob. 7P Ch. 6.5 - Prob. 8P Ch. 6.5 - Prob. 9P Ch. 6.5 - Prob. 10P Ch. 6.5 - Prob. 11P Ch. 6.5 - Prob. 12P Ch. 6.5 - Prob. 13P Ch. 6.5 - Prob. 14P Ch. 6.5 - CAS EXPERIMENT. Variation of a Parameter. (a)...Ch. 6.5 - Prob. 17P Ch. 6.5 - Prob. 18P Ch. 6.5 - Prob. 19P Ch. 6.5 - Prob. 20P Ch. 6.5 - Prob. 21P Ch. 6.5 - Prob. 22P Ch. 6.5 - Prob. 23P Ch. 6.5 - Prob. 24P Ch. 6.5 - Prob. 25P Ch. 6.5 - Prob. 26P Ch. 6.6 - Prob. 2P Ch. 6.6 - Prob. 3P Ch. 6.6 - Prob. 4P Ch. 6.6 - Prob. 5P Ch. 6.6 - Prob. 6P Ch. 6.6 - Prob. 7P Ch. 6.6 - Prob. 8P Ch. 6.6 - Prob. 9P Ch. 6.6 - Prob. 10P Ch. 6.6 - Prob. 11P Ch. 6.6 - Prob. 14P Ch. 6.6 - Prob. 15P Ch. 6.6 - Prob. 16P Ch. 6.6 - Prob. 17P Ch. 6.6 - Prob. 18P Ch. 6.6 - Prob. 19P Ch. 6.6 - Prob. 20P Ch. 6.7 - Prob. 2P Ch. 6.7 - Prob. 3P Ch. 6.7 - Prob. 4P Ch. 6.7 - Prob. 5P Ch. 6.7 - Prob. 6P Ch. 6.7 - Prob. 7P Ch. 6.7 - Prob. 8P Ch. 6.7 - Prob. 9P Ch. 6.7 - Prob. 10P Ch. 6.7 - Prob. 11P Ch. 6.7 - Prob. 12P Ch. 6.7 - Prob. 13P Ch. 6.7 - Prob. 14P Ch. 6.7 - Prob. 15P Ch. 6.7 - Prob. 16P Ch. 6.7 - Prob. 19P Ch. 6.7 - Prob. 20P Ch. 6 - Prob. 1RQ Ch. 6 - Prob. 2RQ Ch. 6 - Prob. 3RQ Ch. 6 - Prob. 4RQ Ch. 6 - Prob. 5RQ Ch. 6 - When and how do you use the unit step function and...Ch. 6 - If you know f(t) = ℒ−1{F(s)}, how would you find...Ch. 6 - Explain the use of the two shifting theorems from...Ch. 6 - Prob. 9RQ Ch. 6 - Prob. 10RQ Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the transform, indicating the method used and...Ch. 6 - Find the inverse transform, indicating the method...Ch. 6 - Prob. 21RQ Ch. 6 - Prob. 22RQ Ch. 6 - Prob. 23RQ Ch. 6 - Prob. 24RQ Ch. 6 - Prob. 25RQ Ch. 6 - Prob. 26RQ Ch. 6 - Prob. 27RQ Ch. 6 - Prob. 28RQ Ch. 6 - Prob. 29RQ Ch. 6 - Prob. 30RQ Ch. 6 - Prob. 31RQ Ch. 6 - Prob. 32RQ Ch. 6 - Prob. 33RQ Ch. 6 - Prob. 34RQ Ch. 6 - Prob. 35RQ Ch. 6 - Prob. 36RQ Ch. 6 - Prob. 37RQ Ch. 6 - Prob. 38RQ Ch. 6 - Prob. 39RQ Ch. 6 - Prob. 40RQ Ch. 6 - Prob. 41RQ Ch. 6 - Prob. 42RQ Ch. 6 - Prob. 43RQ Ch. 6 - Prob. 44RQ Ch. 6 - Prob. 45RQ

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