Chapter 37 - Differential and Difference Equations 4 of 5

What do you do if your P.I is in the same form as your complementary function e.g Complementary function: y = Ae^2x + Be^3x and Particular integral: y = ae^3x

Multiply your particular integral by the independent variable x so your particular integral becomes axe^(3x)

Once you have the solutions to the auxiliary equation how do you get the general solution to the homogenous difference equation?

CASE 1 - ALL DIFFERENT ROOTS If the solutions m₁, m₂, . . . , mₙ of this equation are all distinct, then the general solution: y(x) =α₁e^(m₁x) +α₂e^(m₂x) +···+αₙe^(mₙx) CASE 2 - REPEATED ROOT OF M If a solution m has algebraic multiplicity k, then this particular m contributes in the general solution: y(x) =...(β₁ +β₂x+β₃x² +···+βkx^k−1)e^(mx) +...

State P(D) and Q(x) in the following differential equation 2d³y/dx³ + d²y/dx² - 5dy/dx + 3y = sin(x)

P(D) = 2D³ +D² −5D+3 Q(x) = sin(x)

Give the form of a linear ordinary differential equation with constant coefficients of order n

P(D)y = Q(x) Where Q(x) is a given function and P(D) is a polynomials of the differential operator D = d/dx

How do you form the auxiliary equation? e.g for d²y/dx² + 6dy/dx + 8y = 0

The auxiliary equation would be m² + 6m + 8 = 0 So m₁ = -2 and m₂ = -4 General solution to equation is αe^(-2x) +βe^(-4x)

What do you do if the roots to your auxiliary equation are complex conjugates? i.e m₁ = a + ib and m₂ = a - ib

The real part a becomes the argument of the exponential function and the imaginary part b becomes the argument of the trigonometric functions. Such solutions describe oscillations with amplitude that varies in time.

What is the method for solving a homogenous differential equation?

1) Create auxiliary equation 2) Solve auxiliary equation for m₁ and m₂ 3) General solution to equation is αe^(m₁x) +βe^(m₂x) 4) To find α and β sub in initial conditions

Discuss the behaviour as x → ∞ of the general solution: y(x) = Ae^(-3x) + Be^(2x) + Ce^(3x) + e^(-x)(Dcosx + Esinx)

1) Identify any terms that go to zero 2) Find your dominant term which will be an exponential terms 3) For the dominant term deduce what happens the coefficients are > < = to zero

How do you solve a non-homogeneous differential equation?

1) Solve the homogenous part first called the complementary function 2) Then use a particular integral to find the non-homogeneous part