# How to find parameter functions ?

Is given y = f(x). How to find x = fx(u) and y = fy(u) where u is uniform by length of curve.

Especially I interested in functions:

$f\left(x\right)\ =\ \frac{\sqrt{\left[\left(d\cdot x\right)^{d}\cdot\frac{c^{c}}{\left(d\cdot x+c\right)^{\left(d+c\right)}}\right]}}{x}$ and $f\left(x\right)\ =\ \frac{\left(b-\frac{a}{x\ -c}\right)}{\left(x-c\right)}$

Chat GPT give answer for simple $y= x^2$

```
from sage.symbolic.integration.integral import definite_integral
# Define the variable and the function
u, x = var('u x')
f_x = x^2
# Compute the derivative of x and y with respect to u
df_dx = diff(f_x, x)
# Compute the arc length formula
arc_length = definite_integral(sqrt(df_dx^2 + 1), (x, 0, x))
# Solve for u in terms of s
u_expression = solve(arc_length == u, x)[0].rhs()
# Substitute u back into the original parametric equations
x_u = f_x.subs(x=u_expression)
y_u = u_expression
# Display the results
print("Parametric equations:")
print(f"x = {x_u}")
print(f"y = {y_u}")
but is error - no integration bounds
```

Homework ?