# Solving differential equation , Calculate concentration of sands in water?

$V=$ volume of water in the water tank

$M(t) =$ mass of sands in the water at time $t$

$K(t) =$ Concentration of sands in water at time $t$

$R=$ rate of water flowing out

Concentration sands in water coming in is constant.

If $K(t)$ is the concentration of sands in water flowing out, and $K_{in}$ the concentration flowing in. Then i have this equation $$\frac{dK}{dt}=\frac{R}{V}K_{in} - \frac{R}{V}K$$

How do i solve this differential equation?

If $K_{in} = 10$, what is concentration of sands in the water in long run?

• Do you know about $e^t$? – Michael Galuza Aug 3 '15 at 7:51
• @MichaelGaluza are you talking about properties of Lapace transform? i don't know how i can relate. if not , i don't konw what about et – problematic Aug 3 '15 at 7:54

Notice, $$\frac{dK}{dt}=\frac{R}{V}K_{in}-\frac{R}{V}K$$ $$\implies \frac{dK}{dt}=\frac{R}{V}(K_{in}-K)$$ $$\frac{dK}{K-(K_{in})}=-\frac{R}{V}dt$$ Since, $K_{in}$, concentration sands flowing in, is constant hence integrating both the sides w.r.t. time $t$, we get $$\int\frac{dK}{(K-K_{in})}=-\int \frac{R}{V}dt$$ $$\ln(K-K_{in})=-\frac{R}{V}t+c$$ $$\ln(K-K_{in})-\ln C=-\frac{R}{V}t$$ $$\ln\left(\frac{K-K_{in}}{C}\right)=-\frac{R}{V}t$$ $$\implies K-K_{in}=Ce^{\frac{-R}{V}t}$$ $$\implies K=K_{in}+Ce^{\frac{-R}{V}t}$$
Where $C$ is a constant. Now taking the limit at $t\to \infty$ as follows  the concentration sands in water in long run is given as follows $$\lim_{t\to \infty}K=\lim_{t\to \infty}\left(K_{in}+Ce^{\frac{-R}{V}t}\right)$$ $$K_{\infty}=K_{in}+C(0)$$$$=K_{in}=10$$
• Yes, you will get the same $K$ in long run by taking limit at $t\to \infty$ – Harish Chandra Rajpoot Aug 3 '15 at 16:59
Substitution: $K_{1} = K - K_{in}$. Then equation is $$\frac{d K_1}{d t} = -\frac{R}{V}K_1$$ Solution is $K_1(t) = C * e^{-\frac{R}{V} t}$ where C is constant difined from boundaries i.e. from t=0. Then $$K = K_{in} + C * e^{-\frac{R}{V} t}$$ In long run $K = K_{in}$.
NB: Solution assumes that $K_{in}$ is constant and doesn't depend from t.