CHE 2410 – Mathematical Methods in
Chemical Engineering
CHE 2410
Ordinary Differential Equations
CHE 2410 – Mathematical Methods in
Chemical Engineering
Basic Definition and Terminology
ODE: An ordinary differential equation (ODE) is an
equation that contains ordinary derivatives of one or
more dependent variables with respect to a single
independent variable
PDE: A partial differential equation (PDE) is an
equation that contains partial derivatives of one or more
dependent variables with respect to two or more
independent variables
CHE 2410 – Mathematical Methods in
Chemical Engineering
Classification of ODEs
Ordinary differential equations can be broadly
classified based on:
 Order
 Linearity
CHE 2410 – Mathematical Methods in
Chemical Engineering
Classification of ODEs
Ordinary differential equations can be broadly
classified based on:
 Order:
Order of the highest derivative in the equation
 Linearity
An ordinary differential equation is linear when each
derivative term is linear in y, y', ⋯,y(n) . A linear ODE of
any order can generally be solved by well-established
techniques
CHE 2410 – Mathematical Methods in
Chemical Engineering
Solution Procedure
 Analytical – only certain simple ODEs
can be solved analytically
 Numerical procedure – most commonly
used to solve engineering problems
CHE 2410 – Mathematical Methods in
Chemical Engineering
Separable First Order ODEs
Solution by straight forward integration:
ò
1
dy =
b(y)
ò a(x) dx
CHE 2410 – Mathematical Methods in
Chemical Engineering
Solution by Integrating Factor
Standard form of linear first order ODE:
Sometimes called first order equation with a
forcing function f(x)
Can be solved using Integrating Factor
The general solution:
CHE 2410 – Mathematical Methods in
Chemical Engineering
Exercise 1
Solve the initial value problem:
y
2

y  3x  ;
x
y (1)  5
CHE 2410 – Mathematical Methods in
Chemical Engineering
Exact Solutions
CHE 2410 – Mathematical Methods in
Chemical Engineering
Exercise 2
Solve the differential equation:
2 xy
2



 2 dx  2 x 2 y  4 y dy  0
CHE 2410 – Mathematical Methods in
Chemical Engineering
Forcing Exactness
CHE 2410 – Mathematical Methods in
Chemical Engineering
Exercise 3
Solve the differential equation:
y  
6 xy
4 y  9x2
CHE 2410 – Mathematical Methods in
Chemical Engineering
Homogeneous Function
CHE 2410 – Mathematical Methods in
Chemical Engineering
Exercise 4
Solve the differential equation:
y2
xy   
y
x
CHE 2410 – Mathematical Methods in
Chemical Engineering
Model Formulation
Cooling of a fluid flowing through a circular pipe.
Plug flow: Assume hot fluid is flowing through a pipe
under a radially well-mixed condition. The fluid is
cooled using an external coolant which maintains a
constant wall temperature. Determine how the fluid
temperature changes with pipe length.