Preface

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

Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340
Return to Part III of the course APMA0330

Glossary

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

To check the answer:
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.
  4. Gerald C.F. and Wheatley, P.O., Applied Numerical Analysis, Pearson Education, 7th edition, 2003.
  5. 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

 

Return to Mathematica page
Return to the main page (APMA0330)
Return to the Part 1 (Plotting)
Return to the Part 2 (First Order ODEs)
Return to the Part 3 (Numerical Methods)
Return to the Part 4 (Second and Higher Order ODEs)
Return to the Part 5 (Series and Recurrences)
Return to the Part 6 (Laplace Transform)
Return to the Part 7 (Boundary Value Problems)