Question

Solve the differential equation. dv/ds = (s+1)/(sv + s)

Answer 1

$\frac{dv}{ds} = \frac{s+1}{sv + s}\ \Rightarrow\ \frac{dv}{ds} = \frac{s+1}{s(v+1)}\ \Rightarrow\ (v+1) \frac{dv}{ds} = \frac{s+ 1}{s}\ \Rightarrow \\ \\ (v+1)dv = (1 + \frac{1}{s} )ds\ \Rightarrow\ \int (v+1)dv = \int (1 + \frac{1}{s})ds \Rightarrow \\ \\ \frac{v^2}{2} + v = s + \ln |s| + C\ \Rightarrow\ v^2 + 2v = 2s + 2\ln |s| + 2C$

complete the square

$Complete\ the\ square\\ v^2 + 2v + 1 = 2s + 2\ln |s| + 2C + 1 \ \Rightarrow\ \\ (v+1)^2 = 2s + 2\ln|s| + 2C + 1\ \Rightarrow\ \\ v+1 = \pm \sqrt{ 2s + 2\ln|s| + K} \\ v = -1 \pm \sqrt{ 2s + 2\ln|s| + K},\ \text{where K= 2C + 1 }$