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Math Advanced Math ADV.ENG.MATH (LL) W/WILEYPLUS BUNDLE

The IVP 4 y 1 ′ + y 2 ′ − 2 y 3 ′ = 0 , − 2 y 1 ′ + y 3 ′ = 1 , and 2 y 2 ′ − 4 y 3 ′ = 16 t with initial condition y 1 ( 0 ) = 2 , y 2 ( 0 ) = 0 and y 3 ( 0 ) = 0 . The solution of differential equation is y 1 = 2 + t 2 , y 2 = 2 t _ and y 3 = t + 2 t 2 _ . 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: Take the Laplace transform of the given differential equation 4 y 1 ′ + y 2 ′ − 2 y 3 ′ = 0 . L [ 4 y 1 ′ + y 2 ′ − 2 y 3 ′ ] = L [ 0 ] 4 [ s L ( y 1 ) − y 1 ( 0 ) ] + [ s L ( y 2 ) − y 2 ( 0 ) ] − 2 [ s L ( y 3 ) − y 3 ( 0 ) ] = 0 4 [ s L ( y 1 ) − 2 ] + [ s L ( y 2 ) ] − 2 [ s L ( y 3 ) ] = 0 4 s L ( y 1 ) + s L ( y 2 ) − 2 s L ( y 3 ) = 8 Similarly, take the Laplace transform of the given differential equation − 2 y 1 ′ + y 3 ′ = 1 . L [ − 2 y 1 ′ + y 3 ′ ] = L [ 1 ] − 2 [ s L ( y 1 ) − y 1 ( 0 ) ] + [ s L ( y 3 ) − y 3 ( 0 ) ] = 1 s − 2 [ s L ( y 1 ) − 2 ] + [ s L ( y 3 ) ] = 1 s − 2 s L ( y 1 ) + s L ( y 3 ) = 1 s − 4 Now taking the Laplace transform of the given differential equation 2 y 2 ′ − 4 y 3 ′ = 16 t . L [ 2 y 2 ′ − 4 y 3 ′ ′ ] = − 1 6 L [ t ] 2 [ s L ( y 2 ) − y 2 ( 0 ) ] − 4 [ s L ( y 3 ) − y 3 ( 0 ) ] = − 16 s 2 2 [ s L ( y 2 ) − 0 ] − 4 [ s L ( y 3 ) − 0 ] = − 16 s 2 2 s L ( y 2 ) − 4 s L ( y 3 ) = − 16 s 2 By the use of Crammer rule, the value of L ( y 1 ) is calculated as: L ( y 1 ) = | 8 s − 2 s 1 s − 4 0 s − 16 s 2 2 s − 4 s | | 4 s s − 2 s − 2 s 0 s 0 2 s − 4 s | = 8 ( 0 − 2 s 2 ) − s ( − 4 + 16 s + 16 s ) − 2 s ( 2 − 8 s ) 4 s ( 0 − 2 s 2 ) − s ( 8 s 2 ) − 2 s ( − 4 s 2 ) = − 16 s 2 + 4 s − 16 s 2 − 16 − 4 s + 16 s 2 − 8 s 3 − 8 s 3 + 8 s 3 On the further simplification, the following is obtained. L ( y 1 ) = − 16 s 2 − 16 − 8 s 3 = 2 ( s 2 + 2 ) s 3 = 2 ( 1 s + 1 s 3 ) Now the value of y 1 is calculated as: y 1 = 2 L − 1 ( 1 s + 1 s 3 ) = 2 + t 2 Similarly, find out the value of y 2 = 2 s 2 , y 3 = 1 s 2 + 4 s 3 , and y 2 = 2 t , y 3 = t + 2 t 2 Therefore, the solution of differential equation is y 1 = 2 + t 2 , y 2 = 2 t _ and y 3 = t + 2 t 2 _ .

The IVP 4 y 1 ′ + y 2 ′ − 2 y 3 ′ = 0 , − 2 y 1 ′ + y 3 ′ = 1 , and 2 y 2 ′ − 4 y 3 ′ = 16 t with initial condition y 1 ( 0 ) = 2 , y 2 ( 0 ) = 0 and y 3 ( 0 ) = 0 . The solution of differential equation is y 1 = 2 + t 2 , y 2 = 2 t _ and y 3 = t + 2 t 2 _ . 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: Take the Laplace transform of the given differential equation 4 y 1 ′ + y 2 ′ − 2 y 3 ′ = 0 . L [ 4 y 1 ′ + y 2 ′ − 2 y 3 ′ ] = L [ 0 ] 4 [ s L ( y 1 ) − y 1 ( 0 ) ] + [ s L ( y 2 ) − y 2 ( 0 ) ] − 2 [ s L ( y 3 ) − y 3 ( 0 ) ] = 0 4 [ s L ( y 1 ) − 2 ] + [ s L ( y 2 ) ] − 2 [ s L ( y 3 ) ] = 0 4 s L ( y 1 ) + s L ( y 2 ) − 2 s L ( y 3 ) = 8 Similarly, take the Laplace transform of the given differential equation − 2 y 1 ′ + y 3 ′ = 1 . L [ − 2 y 1 ′ + y 3 ′ ] = L [ 1 ] − 2 [ s L ( y 1 ) − y 1 ( 0 ) ] + [ s L ( y 3 ) − y 3 ( 0 ) ] = 1 s − 2 [ s L ( y 1 ) − 2 ] + [ s L ( y 3 ) ] = 1 s − 2 s L ( y 1 ) + s L ( y 3 ) = 1 s − 4 Now taking the Laplace transform of the given differential equation 2 y 2 ′ − 4 y 3 ′ = 16 t . L [ 2 y 2 ′ − 4 y 3 ′ ′ ] = − 1 6 L [ t ] 2 [ s L ( y 2 ) − y 2 ( 0 ) ] − 4 [ s L ( y 3 ) − y 3 ( 0 ) ] = − 16 s 2 2 [ s L ( y 2 ) − 0 ] − 4 [ s L ( y 3 ) − 0 ] = − 16 s 2 2 s L ( y 2 ) − 4 s L ( y 3 ) = − 16 s 2 By the use of Crammer rule, the value of L ( y 1 ) is calculated as: L ( y 1 ) = | 8 s − 2 s 1 s − 4 0 s − 16 s 2 2 s − 4 s | | 4 s s − 2 s − 2 s 0 s 0 2 s − 4 s | = 8 ( 0 − 2 s 2 ) − s ( − 4 + 16 s + 16 s ) − 2 s ( 2 − 8 s ) 4 s ( 0 − 2 s 2 ) − s ( 8 s 2 ) − 2 s ( − 4 s 2 ) = − 16 s 2 + 4 s − 16 s 2 − 16 − 4 s + 16 s 2 − 8 s 3 − 8 s 3 + 8 s 3 On the further simplification, the following is obtained. L ( y 1 ) = − 16 s 2 − 16 − 8 s 3 = 2 ( s 2 + 2 ) s 3 = 2 ( 1 s + 1 s 3 ) Now the value of y 1 is calculated as: y 1 = 2 L − 1 ( 1 s + 1 s 3 ) = 2 + t 2 Similarly, find out the value of y 2 = 2 s 2 , y 3 = 1 s 2 + 4 s 3 , and y 2 = 2 t , y 3 = t + 2 t 2 Therefore, the solution of differential equation is y 1 = 2 + t 2 , y 2 = 2 t _ and y 3 = t + 2 t 2 _ .

ADV.ENG.MATH (LL) W/WILEYPLUS BUNDLE

ADV.ENG.MATH (LL) W/WILEYPLUS BUNDLE

10th Edition

ISBN: 9781119809210

Author: Kreyszig

Publisher: WILEY

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2. Consider the ODE u' = ƒ (u) = u² + r where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r: (a) Sketch ƒ(u) = u² + r and determine the equilibrium points. (b) Draw the phase line. (d) Determine the stability of the equilibrium points. (d) Plot the direction field and some sample solutions,i.e., u(t) (e) Describe how location of the equilibrium points and their stability change as you increase the parameter r. (f) Using the matlab program phaseline.m generate a solution for each value of r and the initial condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption. (g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to turn in a plot for (g), just describe what happens.

True or False and why

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.

Chapter 6 Solutions

ADV.ENG.MATH (LL) W/WILEYPLUS BUNDLE

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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