J_1(x) = (x^2/10)*(J_1(x) + J_3(x)) How to solve?

[qnach]

Homework Statement

This is not a homewrok

Relevant Equations

J[1, x] = (x^2/10)*(J[1, x] + J[3, x])

Has any one any idea to solve this equation J₁(x) = (x²/10)*(J₁(x) + J₃(x)), in which J are spherical Bessel function normally write as $j_1 (x)$ and $j_3(x)$

Methods 1 serial expansion:
$j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} + \dfrac{(x)^5}{840} + ...$
$j_3 (x) \approx \dfrac{(x^3)}{105}$
I have
$\left(\frac{x}{3}-\frac{x^3}{30} + \frac{x^5}{840} \right) \approx \frac{x^2}{10} \left[ \frac{x}{3} - \frac{x^3}{30} + \frac{x^3}{105} \right]$
This will lead to
$x = \sqrt{ \frac{1 - \sqrt{1 - Y}}{Z} } \approx 0.79$
Y and Z are some complicate expression.

Methods using numerical method.
The result obtained is 2.27 or so....
They are different. Has anyone any idea about what is wrong....

 

[anuttarasammyak]

You forgot to put double dollar signs at head and tail of the equalion lines. I put them.
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Has any one any idea to solve this equation $J[1, x] = (x^2/10)*(J[1, x] + J[3, x])$, in which J are spherical Bessel function normally write as $j_1 (x)$ and $j_3(x)$

Methods 1 serial expansion:
$$j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} + \dfrac{(x)^5}{840} + ...$$
$$j_3 (x) \approx \dfrac{(x^3)}{105}$$
I have
$$\left(\frac{x}{3}-\frac{x^3}{30} + \frac{x^5}{840} \right) \approx \frac{x^2}{10} \left[ \frac{x}{3} - \frac{x^3}{30} + \frac{x^3}{105} \right]$$
This will lead to
$$x = \sqrt{ \frac{1 - \sqrt{1 - Y}}{Z} } \approx 0.79$$
Y and Z are some complicate expression.

Methods using numerical method.
The result obtained is 2.27 or so....
They are different. Has anyone any idea about what is wrong....
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I don't think expansion is a good way. Why don't you express the equation directry by trigonometry fucntion and powers of x by using recurrence formula which reduces $j_3(x)$ to $j_0(x)$ and $j_1(x)$ which is made of trigonometry function and powers of x. I have got a rather simple equation.

[EDIT]

Homework Statement: This is not a homewrok
Relevant Equations: J[1, x] = (x^2/10)*(J[1, x] + J[3, x])

Has any one any idea to solve this equation J₁(x) = (x²/10)*(J₁(x) + J₃(x)), in which J are spherical Bessel function normally write as $j_1 (x)$ and $j_3(x)$

Methods 1 serial expansion:
$j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} + \dfrac{(x)^5}{840} + ...$
$j_3 (x) \approx \dfrac{(x^3)}{105}$
I have
$\left(\frac{x}{3}-\frac{x^3}{30} + \frac{x^5}{840} \right) \approx \frac{x^2}{10} \left[ \frac{x}{3} - \frac{x^3}{30} + \frac{x^3}{105} \right]$
This will lead to
$x = \sqrt{ \frac{1 - \sqrt{1 - Y}}{Z} } \approx 0.79$
Y and Z are some complicate expression.

Double dollar, not a single dollar. Double sharp for ones in text lines.