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Math Advanced Math Custom Kreyszig: Advanced Engineering Mathematics

The value of f ( t ) if the given function L ( f ) = s ( s 2 + 16 ) 2 . The value of f ( t ) is 1 8 t sin 4 t _ . Formula used: The formulas of the inverse Laplace transforms are L − 1 ( s s 2 + a 2 ) = cos a t and L − 1 ( 1 s 2 + a 2 ) = 1 a sin a t . Calculation: The given Laplace function is: L [ f ( t ) ] = s ( s 2 + 16 ) 2 Take the inverse Laplace transform on both sides as follows: f ( t ) = L − 1 [ s ( s 2 + 16 ) 2 ] = L − 1 [ s s 2 + 16 × 1 s 2 + 16 ] Compare the above relation with L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] as shown below: g ( s ) ¯ = s s 2 + 16 and h ( s ) ¯ = 1 s 2 + 16 Now, solve the above equations with Laplace transform: L [ g ( t ) ] = s s 2 + 16 L [ h ( t ) ] = 1 s 2 + 16 Take the inverse Laplace transform on both sides as follows: g ( t ) = L − 1 ( s s 2 + 16 ) = cos 4 t h ( t ) = L − 1 ( 1 s 2 + 16 ) = 1 4 sin 4 t By convolution theorem, L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = g ( t ) h ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ Now, substitute the values g ( t ) = cos 4 t and h ( t ) = 1 4 sin 4 t in L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = ∫ 0 t f ( τ ) g ( t − τ ) d τ . L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = 1 4 ∫ 0 t cos 4 τ sin 4 ( t − τ ) d τ = 1 8 ∫ 0 t 2 cos 4 τ sin 4 ( t − τ ) d τ = 1 8 ∫ 0 t [ sin ( 4 τ + 4 t − 4 τ ) − sin ( 4 τ − 4 t + 4 τ ) ] d τ = 1 8 ∫ 0 t [ sin 4 t − sin ( 8 τ − 4 t ) ] d τ On the further simplification, the following is obtained: L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = 1 8 sin 4 t [ t ] 0 t + 1 8 [ 1 8 cos ( 8 τ − 4 t ) ] 0 t = 1 8 sin 4 t [ t − 0 ] + 1 64 [ cos ( − 4 t ) − cos ( − 4 t ) ] = 1 8 t sin 4 t + 0 = 1 8 t sin 4 t From equation f ( t ) = L − 1 [ s s 2 + 16 × 1 s 2 + 16 ] evaluate f ( t ) as follows: f ( t ) = L − 1 [ s s 2 + 16 × 1 s 2 + 16 ] = L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = 1 8 t sin 4 t Hence, the value of f ( t ) is 1 8 t sin 4 t _ .

The value of f ( t ) if the given function L ( f ) = s ( s 2 + 16 ) 2 . The value of f ( t ) is 1 8 t sin 4 t _ . Formula used: The formulas of the inverse Laplace transforms are L − 1 ( s s 2 + a 2 ) = cos a t and L − 1 ( 1 s 2 + a 2 ) = 1 a sin a t . Calculation: The given Laplace function is: L [ f ( t ) ] = s ( s 2 + 16 ) 2 Take the inverse Laplace transform on both sides as follows: f ( t ) = L − 1 [ s ( s 2 + 16 ) 2 ] = L − 1 [ s s 2 + 16 × 1 s 2 + 16 ] Compare the above relation with L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] as shown below: g ( s ) ¯ = s s 2 + 16 and h ( s ) ¯ = 1 s 2 + 16 Now, solve the above equations with Laplace transform: L [ g ( t ) ] = s s 2 + 16 L [ h ( t ) ] = 1 s 2 + 16 Take the inverse Laplace transform on both sides as follows: g ( t ) = L − 1 ( s s 2 + 16 ) = cos 4 t h ( t ) = L − 1 ( 1 s 2 + 16 ) = 1 4 sin 4 t By convolution theorem, L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = g ( t ) h ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ Now, substitute the values g ( t ) = cos 4 t and h ( t ) = 1 4 sin 4 t in L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = ∫ 0 t f ( τ ) g ( t − τ ) d τ . L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = 1 4 ∫ 0 t cos 4 τ sin 4 ( t − τ ) d τ = 1 8 ∫ 0 t 2 cos 4 τ sin 4 ( t − τ ) d τ = 1 8 ∫ 0 t [ sin ( 4 τ + 4 t − 4 τ ) − sin ( 4 τ − 4 t + 4 τ ) ] d τ = 1 8 ∫ 0 t [ sin 4 t − sin ( 8 τ − 4 t ) ] d τ On the further simplification, the following is obtained: L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = 1 8 sin 4 t [ t ] 0 t + 1 8 [ 1 8 cos ( 8 τ − 4 t ) ] 0 t = 1 8 sin 4 t [ t − 0 ] + 1 64 [ cos ( − 4 t ) − cos ( − 4 t ) ] = 1 8 t sin 4 t + 0 = 1 8 t sin 4 t From equation f ( t ) = L − 1 [ s s 2 + 16 × 1 s 2 + 16 ] evaluate f ( t ) as follows: f ( t ) = L − 1 [ s s 2 + 16 × 1 s 2 + 16 ] = L − 1 [ g ( s ) ¯ ⋅ h ( s ) ¯ ] = 1 8 t sin 4 t Hence, the value of f ( t ) is 1 8 t sin 4 t _ .

Custom Kreyszig: Advanced Engineering Mathematics

Custom Kreyszig: Advanced Engineering Mathematics

10th Edition

ISBN: 9781119166856

Author: Kreyszig

Publisher: JOHN WILEY+SONS INC.CUSTOM

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Chapter 6 Solutions

Custom Kreyszig: 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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