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 .

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!

Worked Example

a) Find the general solution to the differential equation .

8-2-1-particular-integral-a-we-solution

b) Find the general solution to the differential equation .

8-2-1-particular-integral-b-we-solution

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

8-2-1-particular-integral-c-we-solution

 

 

转载自savemyexams

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