What is a second order differential equation?

• A?second order differential equation?is an equation containing second order derivatives (and possibly first order derivatives) but no higher order derivatives
  • For example ?is a second order differential equation
  • And so is
  • But ?is not, because it contains the third order derivative

What are the types of second order differential equation?

• We divide second order differential equations into two main types
• A?homogeneous second order differential equation is of the form ?where?a,?b?and?c?are real constants
  • You may also see this written in the form ?where ?and
• A?non-homogeneous second order differential equation is of the form ?where?a,?b?and?c?are real constants and where f(x) is a non-zero function of?x
  • You may also see this written in the form

How can I solve simple second order differential equations?

• If a second order differential equation is of the form , it will often be possible to solve it simply by using repeated integration
  • A?separate integration constant?will need to be included for each of the integrations
    • This means you will end up with?two integration constants?in your final answer
    • To find the values of these constants you will need two separate initial or boundary conditions
  • See the worked example below for examples of this

What is a complementary function?

• For a second order?homogeneous?differential equation, the equation’s?complementary function?is the general solution to the equation
  • If the differential equation contains initial or boundary conditions you may then use those to find the precise solution to the equation
• For a second order?non-homogeneous?differential equation, the equation’s?complementary function?is only a part of the general solution to the equation
  • For the complete general solution you will need to include the?particular integral?as well (see the following section)
• In order to find a differential equation’s complementary function we use the associated auxiliary equation

What is an auxiliary equation?

• For a second order differential equation ?the associated?auxiliary equation is
  • It is possible that f(x)=0, in which case the differential equation is?homogeneous
  • The auxiliary equation is?exactly the same?whether the differential equation is homogeneous or non-homogeneous
  • The auxiliary equation is a?quadratic equation?in the variable?m
    • The solutions to the auxiliary equation will determine the nature of the associated complementary function

How do I use the auxiliary equation to find the associated complementary function?

• STEP 1: Solve the auxiliary equation ?to find its roots α and β
  • It is possible that the roots will be repeated, with α = β
• STEP 2: The complementary function will be determined by the nature of the roots of the auxiliary equation:
  • CASE 1: ? so that α and β are?distinct real roots
    • The complementary function is ?where?A?and?B?are arbitrary constants
    • This holds even if one of the roots is zero – but if β = 0, say, then the complementary function will become
  • CASE 2:? ?so that there is only?one repeated root?α
    • The complementary function is ?where?A?and?B?are arbitrary constants
  • CASE 3: ? so that α and β are?complex conjugate roots that may be written as
    • The complementary function is ?where?A?and?B?are arbitrary constants
    • Note that if α and β are purely imaginary, then?p?= 0 and the complementary function becomes

How do I solve a second order homogeneous differential equation?

• STEPS 1 & 2: Use the?auxiliary equation?to find the differential equation’s?complementary function?(see above)
  • The complementary function, with its arbitrary constants?A?and?B, is the general solution to the differential equation
• STEP 3: If there are?initial?or?boundary conditions?associated with the differential equation, use these to find the values of the general solution’s arbitrary constants
  • This gives the particular solution to the differential equation with the given initial or boundary conditions
  • Note that because there are two arbitrary constants, you will require two separate initial or boundary conditions to find both constants’ values
  • If the initial or boundary conditions involve ?you will need to differentiate your general solution to find ?in terms of the constants?A?and?B
  • Finding the values of?A?and?B?may require solving simultaneous equations

• Don’t forget to include the complementary function in your solution to a second order non-homogeneous differential equation – the solution is incomplete without it!
• If your attempt to find the constants for a particular integral breaks down and doesn’t appear to have a solution, make sure that you have not used a ‘test form’ of the p.i. that includes terms also found in the complementary function
• Finding the constants for the particular integral can be a very fiddly and algebra-heavy process – be sure to work slowly and methodically to avoid mistakes!