1. SECOND-ORDER DIFFERENTIAL EQUATIONS
PROGRAMME 25
SECOND-ORDER DIFFERENTIAL EQUATIONS

2. Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations

3. Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations

4. Programme 25: Second-order differential equations
Introduction
For any three numbers a, b and c, the two numbers:
are solutions to the quadratic equation:
with the properties:

5. Programme 25: Second-order differential equations
Introduction
The differential equation:
can be re-written to read:
that is:

6. Programme 25: Second-order differential equations
Introduction
The differential equation can again be re-written as:
where:

7. Programme 25: Second-order differential equations
Introduction
The differential equation:
has solution:
This means that:
That is:

8. Programme 25: Second-order differential equations
Introduction
The differential equation:
has solution:
where:

9. Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations

10. Programme 25: Second-order differential equations
Homogeneous equations
The differential equation:
Is a second-order, constant coefficient, linear, homogeneous differential equation. Its solution is found from the solutions to the auxiliary equation:
These are:

11. Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations

12. Programme 25: Second-order differential equations
The auxiliary equation
Real and different roots
Real and equal roots
Complex roots

13. Programme 25: Second-order differential equations
The auxiliary equation
Real and different roots
If the auxiliary equation:
with solution:
where:
then the solution to:

14. Programme 25: Second-order differential equations
The auxiliary equation
Real and equal roots
If the auxiliary equation:
with solution:
where:
then the solution to:

15. Programme 25: Second-order differential equations
The auxiliary equation
Complex roots
If the auxiliary equation:
with solution:
where:
Then the solutions to the auxiliary equation are complex conjugates. That is:

16. Programme 25: Second-order differential equations
The auxiliary equation
Complex roots
Complex roots to the auxiliary equation:
means that the solution of the differential equation:
is of the form:

17. Programme 25: Second-order differential equations
The auxiliary equation
Complex roots
Since:
then:
The solution to the differential equation whose auxiliary equation has complex roots can be written as::

18. Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations

19. Programme 25: Second-order differential equations
Summary
Differential equations of the form:
Auxiliary equation:
Roots real and different: Solution
Roots real and the same: Solution
Roots complex (  j): Solution

20. Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations

21. Programme 25: Second-order differential equations
Inhomogeneous equations
The second-order, constant coefficient, linear, inhomogeneous differential
equation is an equation of the type:
The solution is in two parts y1 + y2:
part 1, y1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation
part 2, y2 is called the particular integral.

22. Programme 25: Second-order differential equations
Inhomogeneous equations
Complementary function
Example, to solve:
Complementary function
Auxiliary equation: m2 – 5m + 6 = 0 solution m = 2, 3
Complementary function y1 = Ae2x + Be3x where:

23. Programme 25: Second-order differential equations
Inhomogeneous equations
Particular integral
(b) Particular integral
Assume a form for y2 as y2 = Cx2 + Dx + E then substitution in:
gives:
yielding:
so that:

24. complementary function + particular integral
Programme 25: Second-order differential equations
Inhomogeneous equations
Complete solution
(c) The complete solution to:
consists of:
complementary function + particular integral
That is:

25. Programme 25: Second-order differential equations
Inhomogeneous equations
Particular integrals
The general form assumed for the particular integral depends upon the form of
the right-hand side of the inhomogeneous equation. The following table can be
used as a guide:

26. Programme 25: Second-order differential equations
Learning outcomes
Use the auxiliary equation to solve certain second-order homogeneous equations
Use the complementary function and the particular integral to solve certain second-order inhomogeneous equations