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

Category: A-level课程, 国际课程, 教材笔记, 福利干货 Date: 2022年10月21日上午9:46

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

8-2-1-2nd-order-diffes-a-we-solution

b) Find the particular solution to the second order differential equation

that satisfies

and

when

8-2-1-2nd-order-diffes-b-we-solution

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)

8-2-1-aux-eqns--compl-functns-a-we-solution

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

8-2-1-aux-eqns--compl-functns-b-we-solution

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!