I have the following direct intergration problem:
dy/dt = v0/(1+kv0t)
Answer
y = 1/k In (1+kv0t) + c
I see where the log comes from, just not sure about the 1/k?
Many thanks in advance.
$\displaystyle \frac{dy}{dt} = \frac{v_0}{1 + kv_0t}$
$\displaystyle y = \int{\frac{v_0}{1 + kv_0t}\,dt}$
$\displaystyle y = \frac{1}{k}\int{\frac{kv_0}{1 + kv_0t}\,dt}$.
Let $\displaystyle u= 1 + kv_0t$ so that $\displaystyle \frac{du}{dt} = kv_0$
So the integral becomes
$\displaystyle y = \frac{1}{k}\int{\frac{1}{u}\,\frac{du}{dt}\,dt}$
$\displaystyle = \frac{1}{k}\int{\frac{1}{u}\,du}$
$\displaystyle = \frac{1}{k}\ln{|u|} + C$
$\displaystyle = \frac{1}{k}\ln{|1 + kv_0t|} + C$.
When you substitute, don't forget to change "dt" also. Let u= 1+ kv0t. The [tex]du= kv_0dt so $\displaystyle dt= \frac{1}{kv_0} du$. Putting that into the integral, $\displaystyle \int \frac{v_0}{1+kv_0t}dt$ becomes $\displaystyle \int \frac{v_0}{u} \frac{1}{kv_0}du= \frac{1}{k}\int \frac{1}{u} du$. The $\displaystyle v_0$ in the numerator cancels the $\displaystyle v_0$ in the denominator but leaves k in the denominator.