# Math Help - Question regard to First Order Differential Equations

1. ## Question regard to First Order Differential Equations

Hey guy, I'm having a problem with First order ODE (again)

The Melting Pot. A tank containing 20,000 litre of blended fuel oil is stirred continuously to keep the mixture homogeneous. The Fuel is blended from a heavy and a light component in equal proportions. The two components are added at a rate of 5 litres/min each, then the blended fuel oil is withdrawn at a rate 10 litres/min to be burnt in a furnace. The blended fuel oil will not burn in the furnace if the heavy component forms more than 80% of the mixture

At a certain instant, the supply of the light component ceases whilst the heavy component continues to be added at the same rate of 5 litres/meter and the homogeneous mixture is withdrawn at the same rate of 10 litres/min.

a) let x(t) be the amount of heavy fuel in the tank (in litres) at time t minutes after the supply of light oil ceases. Show that

$\frac{dx}{dt} + \frac{10*x}{20000 - 5*t} = t, 0<=t<4000$

The only problem in question a) I encounter is the "-5*t" can any one tell me where it comes from?

b) Solve this differential equation for x(t)

This is my working out for this part

I got an answer of $x(t)=(4000-t)^2*5*(4000-t)$

but the answer in the worksheet is

$x(t)=5*(4000-t) - \frac{1}{1600}*(4000-t)^2$

Can any one tell me where I went wrong?

Best Regard
Junks

2. ## Re: Question regard to First Order Differential Equations

If $x$ is the amount of heavy fuel and $y$ is the amount of light fuel at some point in time and these change by $\delta x$ and $\delta y$ in time $\delta t,$ then

$\delta x + \delta y = -5\delta t.$

In the limit as $\delta t \rightarrow 0,$ that gets you $\frac{dx}{dt}+\frac{dy}{dt}=-5.$

Integrate and substitute the initial condition (to evaluate the constant of integration) and you find that $x+y=20000-5t.$

(Actually, on reflection, you can write this down directly.)

Now substitute that into the expression that you should have written down for $\delta x.$

For your solution to the differential equation, you've forgotten to include a constant of integration.