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Chapter 8: Q11-17P (page 460)

Verify (11.18a) and (11.18c) by multiplying by a test function and integrating.

Short Answer

Expert verified

The test function is $x^{2} δ^{'} (x) = 2 δ (x)$ .

Step by step solution

01

Given information

The given expression is

$\int ϕ (x) δ (x) d x$

02

Definition of Integration by Parts

Integration by partsor partial integration is a process that finds theintegralof aproductoffunctionsin terms of the integral of the product of theirderivativeandanti-derivative.

03

Verify the Test Function

Consider

$\int ϕ (x) δ (x) d x = \int (ϕ (x)) δ (x) d x = (x ϕ (x))_{ats = 0}$

Hence $x ϕ (x) = 0$

Consider

$\int_{- \infty}^{\infty} ϕ (x) x^{2} δ (x)^{''} d x = \int_{- \infty}^{\infty} 2 ϕ (x) d x + \int_{- \infty}^{\infty} 4 x ϕ (x)^{'} d x + \int_{- \infty}^{\infty} x^{2} ϕ (x)^{''} d x = {2 ϕ (x)|}_{x = 0} + {4 x ϕ {(x)}^{'}|}_{x = 0} + {x^{2} ϕ {(x)}^{''}|}_{x = 0} = 2 ϕ (0) = \int_{- \infty}^{\infty} ϕ (x) 2 δ (x) d x$

Hence, $x^{2} δ^{'} (x) = 2 δ (x)$

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Most popular questions from this chapter

Use the results which you have obtained in Problems 21 and 22 to find the inverse transform of $(p^{2} + 2 p - 1) I {(p^{2} + 4 p + 5)}^{2}$ .

In Problem 33 to 38, solve the given differential equations by using the principle of superposition [see the solution of equation (6.29)]. For example, in Problem 33, solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus, a polynomial of any degree is kept together in one bracket.

$(D^{2} - 1) y = s i n h x$

If an incompressible fluid flows in a corner bounded by walls meeting at the origin at an angle of 60', the streamlines of the flow satisfy the equation $2 x y d x + (x^{2} - y^{2}) d y = 0$ . Find the streamlines.

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.

$y y' - 2 y^{2} c o t x = s i n x c o s x$

Solve $(12.3)$ if $G = 0$ and $d G / d t = 0$ at $t = 0$ to obtain (12.5). Hint: Use L28 and L3 to find the inverse transform.

See all solutions

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