Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).