Bessel Function: a^2-b^2 Integral Relationship

[Another1]

show that
\(\displaystyle (a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\}\)
when \(\displaystyle J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P)\)

I don, have idea

 

[Another1]

show that
\(\displaystyle (a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\}\)
when \(\displaystyle J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P)\)

(My thinking)
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identities

\(\displaystyle \d{}{x}\left\{ x^{-v}J_{v}(ax)\right\}=-ax^{-v}J_{v+1}(ax)\)
\(\displaystyle \d{}{x}\left\{ x^{v+1}J_{v+1}(ax)\right\}=ax^{v+1}J_{v}(ax) \)

\(\displaystyle \d{}{x}\left\{ x^{v}J_{v}(x)\right\}=x^{v}J_{v-1}(x) \)
\(\displaystyle \d{}{x}\left\{ x^{v+1}J_{v+1}(x)\right\}=x^{v+1}J_{v}(x) \)
\(\displaystyle x^{v+1}J_{v+1}(x)=\int \left\{ x^{v+1}J_{v}(x) \right\} dx \)
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soluion

\(\displaystyle \int J_{v}(ax)J_{v}(bx)x dx = \int \left[ x^{v+1}J_{v}(ax) \right] \left[ x^{-v}J_{v}(bx) \right]dx \)
\(\displaystyle uv-\int v du = \left[ x^{-v}J_{v}(bx) \right] \left[ \frac{x^{v+1}}{a}J_{v+1}(ax) \right]+\frac{b}{a}\int \left[ x^{-v}J_{v+1}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] dx \)

see (by parts again)
\(\displaystyle \int \left[ x^{-v}J_{v+1}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] dx=\int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx \)
\(\displaystyle \int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx=uv-\int vdu \)
\(\displaystyle \int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx=-\frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] dx \)

So...
\(\displaystyle \int J_{v}(ax)J_{v}(bx)x dx = \left[ x^{-v}J_{v}(bx) \right] \left[ \frac{x^{v+1}}{a}J_{v+1}(ax) \right]+\frac{b}{a}\left[ -\frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] \right] dx \)
\(\displaystyle \int J_{v}(ax)J_{v}(bx)x dx = \frac{a}{a^2}\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
\frac{b}{a} \frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a} \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] dx \)
\(\displaystyle a^{2}\int J_{v}(ax)J_{v}(bx)x dx = a\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
b \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + b^2 \int J_{v}(bx) J_{v}(ax) xdx \)
\(\displaystyle (a^{2}-b^{2})\int J_{v}(ax)J_{v}(bx)x dx = a\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
b \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] \)
\(\displaystyle (a^{2}-b^{2})\int J_{v}(ax)J_{v}(bx)x dx = ax J_{v}(bx) J_{v+1}(ax) -
bx J_{v+1}(bx) J_{v}(ax) \)

And finally...
\(\displaystyle (a^{2}-b^{2}) \int_{0}^{P} J_{v}(ax)J_{v}(bx)x \,dx = P(a J_{v}(bP) J_{v+1}(aP) - b J_{v+1}(bP) J_{v}(aP)) \)

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