# Preface

This section gives some examples demonstrated numerical methods used in solving ordinary differential equations of first order.

# Applications

Example:

Example: A 500 liter container initially contains 10 kg of salt. A brine mixture of 100 gramms
of salt per liter is entering the container at 6 liter per minute. The well-mixed contents
are being discharged from the tank at the rate of 6 liters per minute. Express the amount
of salt in the container as a function of time.

Salt is coming at the rate: 6*(0.1)=0.6 kg/min
d yin /dt =0.6 ; d yout/dt = 6x/500

dx/dt = 0.6 -6x/500 ; x(0)=10

DSolve[{x'[t]==6/10-6*x[t]/500, x[0]==10},x[t],t]
Out[15]= {{x[t] -> 10 E^(-3 t/250) (-4 + 5 E^(3 t/250))}}

Suppose that the rate of discharge is reduced to 5 liters per minute.

DSolve[{x'[t]==6/10-5*x[t]/(500+t), x[0]==10},x[t],t]
SolRule[t_] = Apart[x[t] /. First[%]]
Out[1]= {{x[t] -> (1/(
10 (500 + t)^5))(3125000000000000 + 187500000000000 t +
937500000000 t^2 + 2500000000 t^3 + 3750000 t^4 + 3000 t^5 +
t^6)}}
Out[2]= 50 + t/10 - 1250000000000000/(500 + t)^5

Simplify[SolRule'[t] == 6/10 - 5*SolRule[t]/(500 + t)]
Together[SolRule'[t] == Together[6/10 - 5*SolRule[t]/(500 + t)]]
SolRule[0]
Out[3]= True
Out[4]= True
Out[5]= 10

■

1. Bradie, B.A., Friendly Introduction to Numerical Anaysis, Pearson Education,2007
2. Burden, R.L. and Faires, J.D., Numerical Analysis, Cengage Learning, 10 edition, 2015.
3. Gerald C.F. and Wheatley, P.O., Applied Numerical Analysis, Pearson Education, 7th edition, 2003.
4. Hauser, John R., Numerical Methods for Nonlinear Engineering Models, ISBN 978-1-4020-9920-5, 2009, Springer Netherlands, doi: 10.1007/978-1-4020-9920-5