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I don't why al=(2n-1)pi/2. Can you please explain it to me

I don't why al=(2n-1)pi/2. Can you please explain it to me

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I don't why al=(2n-1)pi/2. Can you please explain it to me?Thank you

10:58 АМ Тhu May 6 * 56% AA bartleby.com + M (108) - F G cascade yar... M (38,939) - s... y! Yahoo PM Trigonometri... Solve the bo... = bartleby Q Search for textbooks, step-by-step explanatio... Ask an Expert Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2, 0 &lt; x &lt; L , t &... | Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u dt2,0 &lt; x &lt; L , t &gt; 0 ... DIFFERENTIAL EQUATIONS wth ndary Value Proo Bundle: Differential Equations wit... 9th Edition Dennis G. Zill Publisher: Cengage Learning DENNIS G ZILL Find 2 ISBN: 9781337604901 Chapter 12.7, Problem 6E Textbook Problem Solve the boundary-value problem a? dx2 0 < x < L, t> 0 || dt2 ди u (0, t) = 0, Eª = Fo, t> 0 dx [x=L ди и (х, 0) %3D 0, 0, 0 <х < L. |f=0 The solution u(x, t) represents the longitudinal displacement of a vibrating elastic bar that is anchored at its left end and is subjected to a constant force of magnitude F, at its right end. See Figure 12.4.7 in Exercises 12.4. Eis a constant called the modulus of elasticity. Expert Solution To determine The longitudinal displacement u (x, t) of a vibrating elastic bar such that a² *u = &u_u (0,1) = 0, u (x. 0) = 0, E = Fo and L = 0. dx Ix=L dt Ix=L X GET 10 FREE QUESTIONS Answer to Problem 6E
Transcribed Image Text:10:58 АМ Тhu May 6 * 56% AA bartleby.com + M (108) - F G cascade yar... M (38,939) - s... y! Yahoo PM Trigonometri... Solve the bo... = bartleby Q Search for textbooks, step-by-step explanatio... Ask an Expert Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2, 0 &lt; x &lt; L , t &... | Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u dt2,0 &lt; x &lt; L , t &gt; 0 ... DIFFERENTIAL EQUATIONS wth ndary Value Proo Bundle: Differential Equations wit... 9th Edition Dennis G. Zill Publisher: Cengage Learning DENNIS G ZILL Find 2 ISBN: 9781337604901 Chapter 12.7, Problem 6E Textbook Problem Solve the boundary-value problem a? dx2 0 < x < L, t> 0 || dt2 ди u (0, t) = 0, Eª = Fo, t> 0 dx [x=L ди и (х, 0) %3D 0, 0, 0 <х < L. |f=0 The solution u(x, t) represents the longitudinal displacement of a vibrating elastic bar that is anchored at its left end and is subjected to a constant force of magnitude F, at its right end. See Figure 12.4.7 in Exercises 12.4. Eis a constant called the modulus of elasticity. Expert Solution To determine The longitudinal displacement u (x, t) of a vibrating elastic bar such that a² *u = &u_u (0,1) = 0, u (x. 0) = 0, E = Fo and L = 0. dx Ix=L dt Ix=L X GET 10 FREE QUESTIONS Answer to Problem 6E
10:58 АМ Тhu May 6 * 56% AA bartleby.com + M (108) - F G cascade yar... M (38,939) - s... y! Yahoo PM Trigonometri... Solve the bo... = bartleby Q Search for textbooks, step-by-step explanatio... Ask an Expert Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2 , 0 &lt; x &lt; L , t &... : Solve the boundary-value problem a 2 d 2 u d x 2= d 2 u dt 2,0 &lt; x &lt; L ,t &gt; 0 ... X (x) = c8 sin (ax) (11) Differentiate the above equation with respect to x. X' (x) = ac6cos (ax) Substitute L for x in the above equation. X X' (L) = = ac8cos (aL) ac8cos (aL) = 0 (x' (L) = 0) (2n – 1) 5 (2n – 1) aL a = Therefore, the Eigen values are as follows An = an = ((2n – 1) ) Substitute the value of a in equation (11). X (x) = cg sin ((2n – 1) TX 2L The above equation represents the Eigen functions where. n = 1, 2, 3, .... Substitute a? for 1 in equation (6). T" + a²a?T = 0 The general solution of the above equation is as follows: T (t) = c9cos (aat) + c10sin (aat) (12) Differentiate the above equation with respect to t. T' (t) = -aac9 sin (aat) + aac10 cos (aat) Substitute 0 for t in the above equation. X GET 10 FREE QUESTIONS ) (T' (0) = 0)
Transcribed Image Text:10:58 АМ Тhu May 6 * 56% AA bartleby.com + M (108) - F G cascade yar... M (38,939) - s... y! Yahoo PM Trigonometri... Solve the bo... = bartleby Q Search for textbooks, step-by-step explanatio... Ask an Expert Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2 , 0 &lt; x &lt; L , t &... : Solve the boundary-value problem a 2 d 2 u d x 2= d 2 u dt 2,0 &lt; x &lt; L ,t &gt; 0 ... X (x) = c8 sin (ax) (11) Differentiate the above equation with respect to x. X' (x) = ac6cos (ax) Substitute L for x in the above equation. X X' (L) = = ac8cos (aL) ac8cos (aL) = 0 (x' (L) = 0) (2n – 1) 5 (2n – 1) aL a = Therefore, the Eigen values are as follows An = an = ((2n – 1) ) Substitute the value of a in equation (11). X (x) = cg sin ((2n – 1) TX 2L The above equation represents the Eigen functions where. n = 1, 2, 3, .... Substitute a? for 1 in equation (6). T" + a²a?T = 0 The general solution of the above equation is as follows: T (t) = c9cos (aat) + c10sin (aat) (12) Differentiate the above equation with respect to t. T' (t) = -aac9 sin (aat) + aac10 cos (aat) Substitute 0 for t in the above equation. X GET 10 FREE QUESTIONS ) (T' (0) = 0)
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