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My PDE book does the following:

[tex] \int \phi_x^2 dx [/tex]

Where,

[tex] \phi_x = b-\frac{b}{a} |x| [/tex]

for [tex]|x|> a [/tex] and x=0 otherwise.

Strauss claims:

[tex]\int \phi_x^2 dx = ( \frac{b}{a} ) ^2 2a [/tex]

However, I think there is a mistake. It can be shown that:

[tex] \frac{-3a}{b}(b- \frac{b|x|}{a})^3[/tex] is a Soln. Evaluate this between 0<x<a and you get:

[tex]\frac{b^2 a}{3}[/tex]

Because the absolute value function is symmetric, its twice this value:

[tex]\frac{2b^2 a}{3}[/tex]

Unless I goofed, I think the book is in error.

*Note: Intergration is over the whole real line.

[tex] \int \phi_x^2 dx [/tex]

Where,

[tex] \phi_x = b-\frac{b}{a} |x| [/tex]

for [tex]|x|> a [/tex] and x=0 otherwise.

Strauss claims:

[tex]\int \phi_x^2 dx = ( \frac{b}{a} ) ^2 2a [/tex]

However, I think there is a mistake. It can be shown that:

[tex] \frac{-3a}{b}(b- \frac{b|x|}{a})^3[/tex] is a Soln. Evaluate this between 0<x<a and you get:

[tex]\frac{b^2 a}{3}[/tex]

Because the absolute value function is symmetric, its twice this value:

[tex]\frac{2b^2 a}{3}[/tex]

Unless I goofed, I think the book is in error.

*Note: Intergration is over the whole real line.

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