ODE problem -- Find the amount salt in the tank as water flows through it

[yecko]

Homework Statement

A tank originally contains 100 gal of fresh water. Then water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount salt in the tank at the end of an additional 10 min.

How can the two highlighted part obtain?

Homework Equations

y(t)=(1/μ(t)) ∫ {from to to t} [yoμo+∫(g(t)μ(t))dt]

The Attempt at a Solution

I have tried to substitute the formula, S1(t)=e^(-0.02t) * ∫ {from 0 to t} (S1(t)/50)e^(0.02t) dt
which seems wrong...

Thanks

 

[Orodruin]

Did you try inserting the expression for the integrating factor and checking the equation explicitly? You should find that this gives you back the ODEs for $S_1$ and $S_2$, respectively.

 

[yecko]

S1(t)=e^(-0.02t) * ∫ {from 0 to t} (S1(t)/50)e^(0.02t) dt

integrating factor: e^(0.02t)
but S1(t) and t both have to integrate while I do not actually know what is S1(t), how can I integrate it?
thanks

 

[Orodruin]

Your formula is not correct. It should not contain $S_1(t)$ in the integral. See the general form of the solution in the other thread.