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....
------------

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.

 

[davedyarrow]

This kind of problem with Bessel functions can definitely get confusing, especially when you’re dealing with expressions like J₁(x), J₃(x) and their relations. From similar discussions, the key is usually to apply recursion identities and derivative relations step by step rather than trying to solve everything at once.
Breaking the expression into smaller parts and simplifying each term helps a lot. When I get stuck verifying intermediate steps, I sometimes use la calculadora de alicia to check calculations step by step and avoid algebra mistakes.