# Edexcel A Level Further Maths: Core Pure:复习笔记8.2.1 Solving Second Order Differential Equations

### Second Order Differential Equations

#### 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

#### Worked Example

a) Find the general solution to the second order differential equation .

b) Find the particular solution to the second order differential equation  that satisfies  and  when .

### Auxiliary Equations & Complementary Functions

#### 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

#### Worked Example

a) Find the general solution to each of the following differential equations:
(i)
(ii)
(iii)

b) For each of the differential equations in part (a), find the particular solution that satisfies

### Particular Integral

#### What is a particular integral?

• A particular integral is part of the solution to a second order non-homogeneous differential equation of the form
• When the particular integral is substituted into the left hand side it will produce f(x)
• The other part of the solution is the complementary function associated with the differential equation

#### How do I find the particular integral for a second order non-homogeneous differential equation?

• STEP 1: Choose the correct ‘test form’ of particular integral, based on the function f(x) on the right-hand side of the non-homogeneous differential equation:
 Form of f(x) ‘Test form’ of p.i Notes p λ p is a given constant λ is a constant to be found p + qx λ + μx p & q are given constants λ & μ are constants to be found Use the full test form of the p.i., even if there is no constant term (i.e., even if p = 0) p + qx + rx2 λ + μx + νx2 p, q & r are given constants λ, μ & ν are constants to be found Use the full test form of the p.i., even if there is no constant or x term (i.e., even if p = 0 and/or q =0) pekx λekx k & p are given constants λ is a constant to be found p cos ωx + q sin ωx λ cos ωx + μ sin ωx p, q & ω are given constants λ & μ are constants to be found Use the full test form of the p.i., even if f(x) only contains sin or only contains cos (i.e., even if p = 0 or q =0)
• If all or part of the ‘test form’ of the particular integral occurs in the differential equation’s complementary function, you will need to modify the ‘test form’
• See the following section for how to do this
• STEP 2: Find the first and second derivatives of the ‘test form’ of particular integral
• STEP 3: Substitute the first and second derivatives, along with the ‘test form’ itself, into the differential equation
• STEP 4: By comparing coefficients, determine the correct values to use for the constants in the ‘test form’
• This may require solving simultaneous equations involving the various constants
• The ‘test form’ with the correct values of its constants is the particular integral for the differential equation

#### What if a part of the ‘test form’ of the particular integral already occurs in the differential equation’ complementary function?

• Because the terms of the complementary function are solutions to the homogeneous equation , they cannot also provide possible solutions to the non-homogeneous equation
• If the standard ‘test form’ for the particular integral contains terms that already occur as a part of the complementary function, then the ‘test form’ needs to be modified by adding a factor of x (or possibly of powers of x) to the terms of the ‘test form’
• Some examples:
• The equation  has complementary function
• The standard ‘test form’ of p.i. would be λex, but ex times a constant already occurs in the complementary function
• Therefore  would be used as the ‘test form’ instead
• The equation  has complementary function
• The standard ‘test form’ of p.i. would be λ, but a constant (B) already occurs in the complementary function
• Therefore  would be used as the ‘test form’ instead
• The equation  has complementary function
• The standard ‘test form’ of p.i. would be λe-x, but e-xtimes a constant AND xe-xtimes a constant both already occur in the complementary function
• Therefore  would be used as the ‘test form’ instead

#### How do I solve a second order non-homogeneous differential equation?

• STEP 1: Use the auxiliary equation to find the differential equation’s complementary function (‘c.f.’)
• STEP 2: Find the differential equation’s particular integral (‘p.i.’) including the correct values of any constants
• STEP 3: The general solution to the differential equation is the sum of the complementary function and the particular integral
• I.e., the general solution is y = c.f. + p.i.
• STEP 4: 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 (A and B from the complementary function), 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

#### Exam Tip

• 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!

#### Worked Example

a) Find the general solution to the differential equation .

b) Find the general solution to the differential equation .

c) Find the particular solution to the differential equation  that satisfies .