I don't know if the following question qualifies as research level. If it isn't, sorry.

Set the following terminology:

$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$

$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(x)+c)$

$\beta=\beta(t,x)=t(\ln(x))/2$

$\gamma=\gamma(s,x)=e^{-s\ln(x)/2}$

$c$ is constant.

Now consider the following integral equation:

$$\int_0^\infty \frac{\gamma\sinh(\alpha_1)\cos(\beta)\cos(\alpha_2)}{\sinh(πx)}dx=\int_0^\infty \frac{\gamma\cosh(\alpha_1)\sin(\beta)\sin(\alpha_2)}{\sinh(πx)}dx$$

I want to find the dependancy of $t$ on $s$ explicitly i.e. I want to ask $t=f(s)$ (if possible)

How to achieve this?

(I also asked for the reference for this type of questions earlier)

I don't know if the following question qualifies as research level. If it isn't, sorry.

Set the following terminology:

$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$

$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(x)+c)$

$\beta=\beta(t,x)=t(\ln(x))/2$

$\gamma=\gamma(s,x)=e^{-s\ln(x)/2}$

$c$ is constant.

Now consider the following integral equation:

$$\int_0^\infty \frac{\gamma\sinh(\alpha_1)\cos(\beta)\cos(\alpha_2)}{\sinh(πx)}dx=\int_0^\infty \frac{\gamma\cosh(\alpha_1)\sin(\beta)\sin(\alpha_2)}{\sinh(πx)}dx$$

I want to find the dependancy of $t$ on $s$ explicitly i.e. I want to ask $t=f(s)$ (if possible)

How to achieve this?

I don't know if the following question qualifies as research level. If it isn't, sorry.

Set the following terminology:

$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$

$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(x)+c)$

$\beta=\beta(t,x)=t(\ln(x))/2$

$\gamma=\gamma(s,x)=e^{-s\ln(x)/2}$

$c$ is constant.

Now consider the following integral equation:

$$\int_0^\infty \frac{\gamma\sinh(\alpha_1)\cos(\beta)\cos(\alpha_2)}{\sinh(πx)}dx=\int_0^\infty \frac{\gamma\cosh(\alpha_1)\sin(\beta)\sin(\alpha_2)}{\sinh(πx)}dx$$

I want to find the dependancy of $t$ on $s$ explicitly i.e. I want to ask $t=f(s)$ (if possible)

How to achieve this?

(I also asked for the reference of this type of questions earlier)

I don't know if the following question qualifies as research level. If it isn't, sorry.

Set the following terminology:

$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$

$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(x)+c)$

$\beta=\beta(t,x)=t(\ln(x))/2$

$\gamma=\gamma(s,x)=e^{-s\ln(x)/2}$

$c$ is constant.

Now consider the following integral equation:

$$\int_0^\infty \frac{\gamma\sinh(\alpha_1)\cos(\beta)\cos(\alpha_2)}{\sinh(πx)}dx=\int_0^\infty \frac{\gamma\cosh(\alpha_1)\sin(\beta)\sin(\alpha_2)}{\sinh(πx)}dx$$

I want to find the dependancy of $t$ on $s$ explicitly i.e. I want to ask $t=f(s)$ (if possible)

How to achieve this?

(I also asked for the reference for this type of questions earlier)