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Math Advanced Math ADVANCED ENGINEERING MATHEMATICS

The value of the function f ( t ) if the Laplace transform of f is L ( f ) = ω s 2 ( s 2 + ω 2 ) . The function f ( t ) is f ( t ) = 1 ω 2 ( ω t − sin ω t ) _ . Definition used: The convolution of the functions f and g , denoted by the standard notation f ∗ g and it is defined by the integral h ( t ) = ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ . Theorem used: “If two functions f and g satisfy the assumptions in the existence theorem, so that their transformation F and G exist, the product H = F G is the transform of h given by h ( t ) = ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ ”. Calculation: Given that, the Laplace transform of f is L ( f ) = ω s 2 ( s 2 + ω 2 ) . It can be written as, L ( f ) = ( 1 s 2 ) ⋅ ( ω s 2 + ω 2 ) . Let 1 s 2 be L ( g ) = 1 s 2 and ω s 2 + ω 2 be L ( k ) = ( ω s 2 + ω 2 ) . Thus, the given function can be written as, L ( f ) = L ( g ) ⋅ L ( k ) . Now compute the functions g ( t ) and k ( t ) as follows. Take inverse transform on both sides of L ( g ) = 1 s 2 as follows. L − 1 { L ( g ) } = L − 1 { 1 s 2 } g ( t ) = t Take inverse transform on both sides of L ( k ) = ( ω s 2 + ω 2 ) as follows. L − 1 { L ( k ) } = L − 1 { ω s 2 + ω 2 } k ( t ) = sin ω t Here, it is observed that g ( τ ) = τ and k ( t − τ ) = sin ω ( t − τ ) . Now by the convolution theorem, the function f ( t ) is f ( t ) = ∫ 0 t g ( τ ) k ( t − τ ) d τ . Now compute the function f ( t ) as follows. f ( t ) = ∫ 0 t g ( τ ) k ( t − τ ) d τ = ∫ 0 t τ sin ω ( t − τ ) d τ Use integration by parts and obtain f ( t ) as follows. Here, u = τ and v ′ = sin ω ( t − τ ) . Then the integral becomes, ∫ u d v = u v − ∫ v d u . That is, f ( t ) = [ τ ( − cos ω ( t − τ ) − ω ) ] 0 t − ∫ 0 t ( − cos ω ( t − τ ) − ω ) d τ = [ τ ( cos ω ( t − τ ) ω ) ] 0 t − ∫ 0 t ( cos ω ( t − τ ) ω ) d τ = 1 ω [ [ τ ( cos ω ( t − τ ) ) ] 0 t − ∫ 0 t ( cos ω ( t − τ ) ) d τ ] = 1 ω [ [ τ ( cos ω ( t − τ ) ) ] 0 t − [ ( sin ω ( t − τ ) ) − ω ] 0 t ] Simplify further as follows. f ( t ) = 1 ω [ [ τ ( cos ω ( t − τ ) ) ] 0 t − [ ( sin ω ( t − τ ) ) − ω ] 0 t ] = 1 ω [ [ t ( cos ω ( t − t ) ) − 0 ] + [ ( sin ω ( t − t ) ) ω − ( sin ω ( t − 0 ) ) ω ] ] = 1 ω [ [ t ( 1 ) ] + [ 0 − ( sin ω t ) ω ] ] = 1 ω [ t − sin ω t ω ] On further simplification. f ( t ) = 1 ω [ t − sin ω t ω ] = 1 ω 2 [ ω t − sin ω t ] = 1 ω 2 ( ω t − sin ω t ) Thus, the function f ( t ) is f ( t ) = 1 ω 2 ( ω t − sin ω t ) _ .

The value of the function f ( t ) if the Laplace transform of f is L ( f ) = ω s 2 ( s 2 + ω 2 ) . The function f ( t ) is f ( t ) = 1 ω 2 ( ω t − sin ω t ) _ . Definition used: The convolution of the functions f and g , denoted by the standard notation f ∗ g and it is defined by the integral h ( t ) = ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ . Theorem used: “If two functions f and g satisfy the assumptions in the existence theorem, so that their transformation F and G exist, the product H = F G is the transform of h given by h ( t ) = ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ ”. Calculation: Given that, the Laplace transform of f is L ( f ) = ω s 2 ( s 2 + ω 2 ) . It can be written as, L ( f ) = ( 1 s 2 ) ⋅ ( ω s 2 + ω 2 ) . Let 1 s 2 be L ( g ) = 1 s 2 and ω s 2 + ω 2 be L ( k ) = ( ω s 2 + ω 2 ) . Thus, the given function can be written as, L ( f ) = L ( g ) ⋅ L ( k ) . Now compute the functions g ( t ) and k ( t ) as follows. Take inverse transform on both sides of L ( g ) = 1 s 2 as follows. L − 1 { L ( g ) } = L − 1 { 1 s 2 } g ( t ) = t Take inverse transform on both sides of L ( k ) = ( ω s 2 + ω 2 ) as follows. L − 1 { L ( k ) } = L − 1 { ω s 2 + ω 2 } k ( t ) = sin ω t Here, it is observed that g ( τ ) = τ and k ( t − τ ) = sin ω ( t − τ ) . Now by the convolution theorem, the function f ( t ) is f ( t ) = ∫ 0 t g ( τ ) k ( t − τ ) d τ . Now compute the function f ( t ) as follows. f ( t ) = ∫ 0 t g ( τ ) k ( t − τ ) d τ = ∫ 0 t τ sin ω ( t − τ ) d τ Use integration by parts and obtain f ( t ) as follows. Here, u = τ and v ′ = sin ω ( t − τ ) . Then the integral becomes, ∫ u d v = u v − ∫ v d u . That is, f ( t ) = [ τ ( − cos ω ( t − τ ) − ω ) ] 0 t − ∫ 0 t ( − cos ω ( t − τ ) − ω ) d τ = [ τ ( cos ω ( t − τ ) ω ) ] 0 t − ∫ 0 t ( cos ω ( t − τ ) ω ) d τ = 1 ω [ [ τ ( cos ω ( t − τ ) ) ] 0 t − ∫ 0 t ( cos ω ( t − τ ) ) d τ ] = 1 ω [ [ τ ( cos ω ( t − τ ) ) ] 0 t − [ ( sin ω ( t − τ ) ) − ω ] 0 t ] Simplify further as follows. f ( t ) = 1 ω [ [ τ ( cos ω ( t − τ ) ) ] 0 t − [ ( sin ω ( t − τ ) ) − ω ] 0 t ] = 1 ω [ [ t ( cos ω ( t − t ) ) − 0 ] + [ ( sin ω ( t − t ) ) ω − ( sin ω ( t − 0 ) ) ω ] ] = 1 ω [ [ t ( 1 ) ] + [ 0 − ( sin ω t ) ω ] ] = 1 ω [ t − sin ω t ω ] On further simplification. f ( t ) = 1 ω [ t − sin ω t ω ] = 1 ω 2 [ ω t − sin ω t ] = 1 ω 2 ( ω t − sin ω t ) Thus, the function f ( t ) is f ( t ) = 1 ω 2 ( ω t − sin ω t ) _ .

ADVANCED ENGINEERING MATHEMATICS

ADVANCED ENGINEERING MATHEMATICS

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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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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