Module 6: Second Order Differential Equations

Flashcards generated from lecture notes on Second Order Differential Equations.

What is a second order differential equation?

An equation for an unknown function 'y' that contains the second derivative of y, but no higher derivatives.

What is the general form of a linear second order differential equation?

P(x)y" + Q(x)y' + R(x)y = G(x)

What is the form of the general solution to a linear second order homogeneous DE?

y(x) = c₁y₁(x) + c₂y₂(x), where y₁ & y₂ are linearly independent solutions of the DE.

What is the condition for two functions y₁ and y₂ to be linearly dependent on an interval x ∈ (a, b)?

k₁y₁(x) + k₂y₂(x) = 0 for all x ∈ (a, b), where k₁ ≠ 0 and k₂ ≠ 0.

If y₁ and y₂ are solutions to y'' + p(x)y' + q(x)y = 0, when are y₁ & y₂ linearly independent?

Wy₁, y₂ = y₁y₂' - y₂y₁' ≠ 0 for all x ∈ (a, b).

How can linearly dependent functions be easily recognized?

Functions that are scalar multiples of one another.

What is the form of a homogeneous differential equation with constant coefficients?

ay'' + by' + cy = 0

What form of solution is used to solve a second order, homogeneous, constant coefficient differential equation?

y = e^(λx) is a solution to ay'' + by' + cy = 0.

What is the auxiliary equation for ay'' + by' + cy = 0?

ax² + bx + c = 0

If the auxiliary equation has two real roots λ₁ and λ₂, what is the general solution to the DE?

y(x) = c₁e^(λ₁x) + c₂e^(λ₂x)

If the auxiliary equation has one real root λ, what is the general solution to the DE?

y(x) = c₁e^(λx) + c₂xe^(λx)

If the auxiliary equation has complex conjugate roots λ = α ± βi, what is the general solution to the DE?

y(x) = c₁e^(αx)cos(βx) + c₂e^(αx)sin(βx)

What is required for an initial value problem (IVP) for a second order DE?

Requires the value of y and its first derivative y' to be specified for some value of x.

What is the form of a non-homogeneous second order DE with constant coefficients?

ay'' + by' + cy = G(x)

What is the general solution of the non-homogeneous DE ay'' + by' + cy = G(x)?

y(x) = yc(x) + yp(x)

When can the method of undetermined coefficients be applied to find a particular solution?

A polynomial, an exponential of the form e^(kx), a cos or sin function of the form cos(kx) or sin(kx), or a product of any of the above functions.

If G(x) is a polynomial of order n, what form should the trial solution yp(x) take?

yp(x) = An x^n + A(n-1) x^(n-1) + … + A₀

If G(x) = Ce^(kx), what form should the trial solution yp(x) take?

yp(x) = Ae^(kx)

If G(x) = Ccos(kx) or G(x) = Csin(kx), what form should the trial solution yp(x) take?

yp(x) = Acos(kx) + Bsin(kx)

If G(x) is a product of multiple functions, G(x) = G₁(x) * G₂(x) * G₃(x) …, how you construct the trial solution?

yp(x) = yp₁(x) * yp₂ (x) * yp₃(x) …

What adjustment must be made to the trial solution yp if it contains terms that are also present in the complementary solution yc?

If any term in the trial solution yp is also in the complementary solution yc, then multiply yp by x. Repeat if necessary.

When solving an IVP, when should you solve for the constants c₁ and c₂?

Solve for the constants c₁ and c₂ using the initial conditions after including the particular solution yp into the general solution.

What is the form of yp(x) in the method of variation of parameters, given that y₁(x) and y₂(x) are linearly independent solutions to the complementary equation?

yp(x) = u₁(x)y₁(x) + u₂(x)y₂(x)

In the method of variation of parameters, what is the expression for du₁/dx?

du₁/dx = -y₂(x)G(x) / Wy₁, y₂

In the method of variation of parameters, what is the expression for du₂/dx?

du₂/dx = y₁(x)G(x) / Wy₁, y₂

What is the differential equation that models the vibration of a spring with mass m, damping constant c, and spring constant k?

m(d²x/dt²) + c(dx/dt) + kx = 0

What are the three cases that arise when solving the auxiliary equation for a damped spring system?

Over damping, critical damping, and under damping.

What is the differential equation that models the vibration of a spring affected by an external force F(t)?

m(d²x/dt²) + c(dx/dt) + kx = F(t)

What is resonance in the context of forced vibrations, and what effect does it have in the absence of damping?

Resonance occurs when the forcing frequency matches the natural resonant frequency of the spring, leading to continuous growth in the magnitude of oscillations if damping is absent.