I need this question completed in 5 minutes with handwritten working out 4. Consider the boundary value problem (BVP) for the unknown u(x),du(e-xdu) = = x, u(0) = 0, (1) = 0.dxWrite down the BVP that the corresponding Green's function should(a)satisfy.(b)ddxFind the Green's function

Given differential equation is Here first we consider associated homogeneous problem to check…

Answered: 4. Consider the boundary value problem…

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 need this question completed in 5 minutes with handwritten working out

4. Consider the boundary value problem (BVP) for the unknown u(x),
du
(e-xdu) = = x, u(0) = 0, (1) = 0.
dx
Write down the BVP that the corresponding Green's function should
(a)
satisfy.
(b)
d
dx
Find the Green's function

Expert Solution

Step 1: Green's function construction

Given differential equation is $fraction numerator d over denominator d x end fraction open parentheses e to the power of negative x end exponent fraction numerator d u over denominator d x end fraction close parentheses equals x equals f left parenthesis x right parenthesis space left parenthesis s a y right parenthesis comma u left parenthesis 0 right parenthesis equals 0 comma u apostrophe left parenthesis 1 right parenthesis equals 0$

Here first we consider associated homogeneous problem $fraction numerator d over denominator d x end fraction open parentheses e to the power of negative x end exponent fraction numerator d u over denominator d x end fraction close parentheses equals 0 comma u left parenthesis 0 right parenthesis equals 0 comma u apostrophe left parenthesis 1 right parenthesis equals 0$ to check Green's function exists or not.

it's general solution become $u left parenthesis x right parenthesis equals A e to the power of x plus B$

Now $u left parenthesis 0 right parenthesis equals 0 rightwards double arrow A plus B equals 0 space a n d space u apostrophe left parenthesis 1 right parenthesis equals 0 rightwards double arrow A equals 0$

solving we get $B equals 0$

so given SL condition gives trivial solution here.

Here two linearly independent solutions are $1 comma e to the power of x$