Differential Equations Mixing Problem (REVISED)

[cmajor47]

1. Suppose we have two tanks, each with an inflow pipe and an outflow pipe and also connected to each other by pipes. At the start, tank A contains 40 gallons of clean water and Tank B contains 25 gallons of clean water. At t = 0 brine containing 0.5 lbs of salt per gallon begins to flow into tank A at the rate of 5 gallons per minute. The well stirred mixture flows into tank B at the rate of 3 gallons per minute. Brine containing 1 lb of salt per gallon flows into tank B at the rate of 1 gallon per minute. The resulting brine flows from tank B into tank A at the rate of 1 gallon per minute. The well-stirred mixture flows from tank A to the drain at the rate of 3 gallons per minute and the solution in tank B flows to the drain at the rate of 3 gallons per minute. Find expressions for the amount of salt in each tank at time t.

Homework Equations

We need to solve this problem using matrices, eigen values, eigen vectors, etc.
Let x(t) be the amount of salt in tank A and y(t) be the amount of salt in tank B.

The Attempt at a Solution

I know that my work is right up to a certaing point. Here is what I've got:
$$\frac{dx}{dt}$$=2.5 - $$\frac{3x}{40}$$ - $$\frac{3x}{40}$$ + $$\frac{y}{25}$$

and that $$\frac{dy}{dt}$$=1 + $$\frac{3x}{40}$$ - $$\frac{y}{25}$$ - $$\frac{3y}{25}$$


So $$\frac{dx}{dt}$$=- $$\frac{3x}{20}$$ + $$\frac{y}{25}$$ + 2.5

and $$\frac{dy}{dt}$$= $$\frac{3x}{40}$$ - $$\frac{4y}{25}$$ + 1

The rest of the work I've attached in pictures because it's too long to type out.

In class my professor told us that we should get x'=-2c₁e^-0.21t+2c₂e^-0.1t+20.95 and
y'=3c₁e^-0.21t+5c₂e^-0.1t+16.07

I can't figure out how to find the 20.95 and 16.07 that are added onto the end of x' and y'.

Can anyone help with this?

 

[kurt.physics]

Thanks!To find the constants c1 and c2, you need to use the initial conditions. At t=0, we have x(0)=40 and y(0)=25. Plugging these values into the solutions for x' and y', we get:-2c1 + 2c2 + 20.95 = 40 3c1 + 5c2 + 16.07 = 25Solving this system of equations, we get c1 = 10 and c2 = 15. Therefore, the final solutions are:x' = -20e-0.21t + 30e-0.1t + 20.95y' = 30e-0.21t + 75e-0.1t + 16.07