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The value of the function f ( t ) if the Laplace transform of f is L ( f ) = e − a s s ( s − 2 ) . The function f ( t ) is f ( t ) = { 1 2 ( e 2 ( t − a ) − 1 ) if t > a 0 if t < a _ . 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 ) = e − a s s ( s − 2 ) . It can be written as, L ( f ) = ( e − a s s ) ⋅ ( 1 s − 2 ) . Let e − a s s be L ( g ) = e − a s s and 1 s − 2 be L ( k ) = ( 1 s − 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 ) = e − a s s as follows. L − 1 { L ( g ) } = L − 1 { e − a s s } g ( t ) = u ( t − a ) Here, the function u ( t − a ) is an unit step function and it is defined as, u ( t − a ) = { 0 if t < a 1 if t > a . Take inverse transform on both sides of L ( k ) = ( 1 s − 2 ) as follows. L − 1 { L ( k ) } = L − 1 { 1 s − 2 } k ( t ) = e 2 t Here, it is observed that g ( τ ) = { 0 if t < a 1 if t > a and k ( t − τ ) = e 2 ( 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 a g ( τ ) k ( t − τ ) d τ + ∫ a t g ( τ ) k ( t − τ ) d τ = ∫ 0 a 0 ⋅ e 2 ( t − τ ) d τ + ∫ a t 1 ⋅ e 2 ( t − τ ) d τ = 0 + ∫ a t e 2 ( t − τ ) d τ Simplify further as follows. f ( t ) = ∫ a t e 2 t − 2 τ d τ = e 2 t [ e − 2 τ 2 ] a t = e 2 t 2 [ e − 2 t − e − 2 a ] = 1 2 ( e 2 ( t − a ) − 1 ) Thus, the function f ( t ) is f ( t ) = { 1 2 ( e 2 ( t − a ) − 1 ) if t > a 0 if t < a _ .

The value of the function f ( t ) if the Laplace transform of f is L ( f ) = e − a s s ( s − 2 ) . The function f ( t ) is f ( t ) = { 1 2 ( e 2 ( t − a ) − 1 ) if t > a 0 if t < a _ . 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 ) = e − a s s ( s − 2 ) . It can be written as, L ( f ) = ( e − a s s ) ⋅ ( 1 s − 2 ) . Let e − a s s be L ( g ) = e − a s s and 1 s − 2 be L ( k ) = ( 1 s − 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 ) = e − a s s as follows. L − 1 { L ( g ) } = L − 1 { e − a s s } g ( t ) = u ( t − a ) Here, the function u ( t − a ) is an unit step function and it is defined as, u ( t − a ) = { 0 if t < a 1 if t > a . Take inverse transform on both sides of L ( k ) = ( 1 s − 2 ) as follows. L − 1 { L ( k ) } = L − 1 { 1 s − 2 } k ( t ) = e 2 t Here, it is observed that g ( τ ) = { 0 if t < a 1 if t > a and k ( t − τ ) = e 2 ( 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 a g ( τ ) k ( t − τ ) d τ + ∫ a t g ( τ ) k ( t − τ ) d τ = ∫ 0 a 0 ⋅ e 2 ( t − τ ) d τ + ∫ a t 1 ⋅ e 2 ( t − τ ) d τ = 0 + ∫ a t e 2 ( t − τ ) d τ Simplify further as follows. f ( t ) = ∫ a t e 2 t − 2 τ d τ = e 2 t [ e − 2 τ 2 ] a t = e 2 t 2 [ e − 2 t − e − 2 a ] = 1 2 ( e 2 ( t − a ) − 1 ) Thus, the function f ( t ) is f ( t ) = { 1 2 ( e 2 ( t − a ) − 1 ) if t > a 0 if t < a _ .

Advanced Engineering Mathematics

Advanced Engineering Mathematics

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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9. Needing a break from studying, you take a walk to the Pogonip koi pond, whereupon a wild-eyed stranger pops out from behind a redwood tree and directs the following polemic in your general direction: "The lies those so-called teachers at that university promulgate, let me tell you. I know the truth that they don't want you to know. As plain as day, " = 0 for all n ≥0. It's an easy induction proof, see?" He hands you a leaflet, where you see the proof that they don't want you to see: We proceed by strong induction on n. Base case: n = 0. We have 10: Induction step: Assume that d1 = = = 0. dx dxk dx = 0 for all kn. Then, by the product rule, nd dx da 1x+1 = 1/1(x²x²) = x²±²x² + x 11 x² d = x.0+x¹.0 0. dx This completes the induction. That derivative rule doesn't seem like the one you learned, but there's nothing obviously wrong with the proof. Is he right, are the math professors propping up the interests of Big Calculus? Or should he have paid better attention in CSE 16? What's going…

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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