(a) Explanation Consider the given differential equation d d r [ 1 r d d r ( r d w d r ) ] = q 2 D r . Use the product rule d d r ( u v ) = u d v d r + v d u d r twice for differentiating the left hand side of the above function. d d r [ 1 r d d r ( r d w d r ) ] = d d r [ 1 r ( r d 2 w d r 2 + d w d r ) ] = d d r [ d (b) To determine To solve: The equation d d r [ 1 r d d r ( r d w d r ) ] = q 2 D r by integrating only with respect to r and to show the result is equivalent to the solution w ( r ) = c 1 + c 2 ln r + c 3 r 2 + q 64 D r 4 .

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.7, Problem 48E

(a)

To determine

To verify: The equation $d3wdr3+1rd2wdr2−1rdwdr=q2Dr$ can be written in the alternate form $ddr[1rddr(rdwdr)]=q2Dr$ .

(b)

To determine

To solve: The equation $ddr[1rddr(rdwdr)]=q2Dr$ by integrating only with respect to r and to show the result is equivalent to the solution $w(r)=c1+c2lnr+c3r2+q64Dr4$ .

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Chapter 4.7, Problem 47E

Chapter 4.8, Problem 1E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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To verify: The equation d 3 w d r 3 + 1 r d 2 w d r 2 − 1 r d w d r = q 2 D r can be written in the alternate form d d r [ 1 r d d r ( r d w d r ) ] = q 2 D r .

Solution Summary: The author explains the alternative form of the equation drleft, which can be written in the alternate form.