How do you find the general solution of the differential equation?

Rewrite as Integral of dx/x(5- x) = integral of k dr Now we have to put 1/x(5 - x) in partial fractions and we get 1/5( 1/x + 1/(5 - x))

Integrating we get 1/5(lnx - ln(5 -x )) = kr + C.

Given, dx / dr = k x(5 - x)
=> dx / x(5 - x) = k dr
=> {1 / x(5 - x)} dx = k dr
=> (1/5){(1/x) + 1/(5 - x)} dx = k dr ; Integrating both sides
=> (1/5){ln x - ln (5 - x)} = k r - (1/5)(ln A) [ adding the integration constant in the form - (1/5)(ln A) ]
=> (1/5) ln{x / (5 - x)} + (1/5)(ln A) = k r
=> (1/5)[ln{x / (5 - x)} + ln A] = kr
=> ln{Ax / (5 - x)} = 5kr
=> {Ax / (5 - x)} = e^(5kr)

dx/dr = kx(5 - x)

dx/x(5 - x) = dr

1/5*[ln|x| - ln|5 - x|] = r + C

solve for x.