Explanation Given: The temperature within the fin is represented as T ( r ) and the constant air temperature is T m = 70 . The internal and external radius of the fin is r 0 = 1 , r 1 = 3 and the relation is w ( r ) = T ( r ) − 70 . Calculation: The differential equation is given as follows: d d r ( r d T d r ) = a 2 r ( T − T m ) … … ( 1 ) Simplify the equation ( 1 ) . d d r ( r d T d r ) = a 2 r ( T − T m ) r d 2 T d r 2 + d T d r = a 2 r ( T − T m ) r d 2 T d r 2 + d T d r − a 2 r ( T − T m ) = 0 Multiply the above equation by r . r 2 d 2 T d r 2 + r d T d r − a 2 r 2 ( T − T m ) = 0 Substitute 70 for T m in the above equation. r 2 d 2 T d r 2 + r d T d r − a 2 r 2 ( T − 70 ) = 0 Substitute T ( r ) = w ( r ) + 70 in the above equation. r 2 d 2 ( w ( r ) + 70 ) d r 2 + r d ( w ( r ) + 70 ) d r − a 2 r 2 ( w ( r ) ) = 0 r 2 d 2 w d r 2 + r d w d r − a 2 r 2 w = 0 Label the above equation as follows: r 2 d 2 w d r 2 + r d w d r − a 2 r 2 w = 0 … … ( 2 ) Equation ( 2 ) is in the form of modified Bessel’s equation of order ν = 0 . The general solution of equation ( 2 ) is, as follows: w ( r ) = c 1 I 0 ( a r ) + c 2 K 0 ( a r ) Substitute the expression for w ( r ) in the equation T ( r ) = w ( r ) + 70 . T ( r ) = 70 + c 1 I 0 ( a r ) + c 2 K 0 ( a r ) … … ( 3 ) Substitute 1 for r in equation ( 3 ) . T ( 1 ) = 70 + c 1 I 0 ( a ) + c 2 K 0 ( a ) 70 + c 1 I 0 ( a ) + c 2 K 0 ( a ) = 160 c 1 I 0 ( a ) + c 2 K 0 ( a ) = 160 − 70 ( T ( 1 ) = 70 ) c 1 I 0 ( a ) + c 2 K 0 ( a ) = 90 Label the above equation as equation ( 4 ) . c 1 I 0 ( a ) + c 2 K 0 ( a ) = 90 … … ( 4 ) Substitute 3 for r in equation ( 3 )

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 6, Problem 28RE

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The solution of the differential equation ddr(rdTdr)=a2r(T−Tm) such that T(1)=160 and T(3)=90.

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Chapter 6, Problem 27RE

Chapter 7.1, Problem 1E

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Chapter 6 Solutions

Differential Equations with Boundary-Value Problems

Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 10E

Ch. 6.1 - Prob. 11E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 13E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 16E Ch. 6.1 - In Problems 17 and 18 use an appropriate series in...Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - Prob. 28E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - In Problems 3538 proceed as in Example 4 and find...Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.2 - In Problems 1 and 2 without actually solving the...Ch. 6.2 - Prob. 2E Ch. 6.2 - In Problems 3–6 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - Prob. 9E Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Prob. 13E Ch. 6.2 - Prob. 14E Ch. 6.2 - Prob. 15E Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Prob. 18E Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Prob. 23E Ch. 6.2 - In Problems 23 and 24 use the procedure in Example...Ch. 6.2 - Without actually solving the differential equation...Ch. 6.2 - How can the power series method be used to solve...Ch. 6.2 - Prob. 27E Ch. 6.2 - Prob. 28E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 9E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 11E Ch. 6.3 - Prob. 12E Ch. 6.3 - In Problems 13 and 14, x = 0 is a regular singular...Ch. 6.3 - Prob. 14E Ch. 6.3 - In Problems 1524, x = 0 is a regular singular...Ch. 6.3 - Prob. 16E Ch. 6.3 - Prob. 17E Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Prob. 23E Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - In Problems 1320 use (20) to find the general...Ch. 6.4 - Prob. 21E Ch. 6.4 - Assume that b in equation (20) can be pure...Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Use the recurrence relation in Problem 28 along...Ch. 6.4 - Prob. 33E Ch. 6.4 - Prob. 34E Ch. 6.4 - Use the change of variables s=2kmet/2 to show that...Ch. 6.4 - Prob. 36E Ch. 6.4 - Use the result in parts (a) and (b) of Problem 36...Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - (a) Use the explicit solutions y1(x) and y2(x) of...Ch. 6.4 - Prob. 47E Ch. 6.4 - Prob. 48E Ch. 6.4 - Find the first three positive values of for which...Ch. 6.4 - The differential equation y 2xy + 2y = 0 is known...Ch. 6.4 - (a) When = n is a nonnegative integer, Hermites...Ch. 6.4 - Prob. 55E Ch. 6.4 - Prob. 56E Ch. 6 - In Problems 1 and 2 answer true or false without...Ch. 6 - Prob. 2RE Ch. 6 - Prob. 3RE Ch. 6 - Prob. 4RE Ch. 6 - Prob. 5RE Ch. 6 - Prob. 6RE Ch. 6 - Prob. 7RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 9RE Ch. 6 - Prob. 10RE Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 16RE Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 20RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Prob. 24RE Ch. 6 - Prob. 25RE Ch. 6 - Express the general solution of the given...Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE

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The solution of the differential equation d d r ( r d T d r ) = a 2 r ( T − T m ) such that T ( 1 ) = 160 and T ( 3 ) = 90 .

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The solution of the differential equation ddr(rdTdr)=a2r(T−Tm) such that T(1)=160 and T(3)=90.

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Chapter 6 Solutions