Question

Solve the differential equation.

dv/ds = (s+1)/(sv + s)

Answer 1

`(dv)/(ds) = (s+1)/(sv + s)\ \Rightarrow\ (dv)/(ds) = (s+1)/(s(v+1))\ \Rightarrow\ (v+1) (dv)/(ds) = (s+ 1)/(s)\ \Rightarrow \\ \\ (v+1)dv = (1 + (1)/(s) )ds\ \Rightarrow\ \int (v+1)dv = \int (1 + (1)/(s))ds \Rightarrow \\ \\ (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 √( 2s + 2\ln|s| + K) \\ v = -1 \pm √( 2s + 2\ln|s| + K),\ \text{where $K= 2C + 1$}`