
Upon which principle does reduction of order lie?
Given a solution to a differential equation y_{1}, the second solution, y_{2} can be found by multiplying y_{1 }by a function u(x).

What formula is useful for reduction of order?
 y_{2}(x) = y_{1}(x)∫ e^{∫P(x) dx} y_{12}^{ }(x) dx
 (Given y" + P(x)y' + Q(x)y = 0)

For an equation in the form Iy" + Jy' + Ky = sin(x), what form of solution should be proposed for y_{p} if using the method of undetermined coefficients?
Acos(x) + Bsin(x)

What is the basic equation format of variation of parameters?
y_{p} = u_{1}y_{1} + u_{2}y_{2}

In variation of parameters, how do you find u_{1} and u_{2}?
u_{1}' = W_{1}/W, u_{2} = W_{2}/W. Then integrate to find u_{1} and u_{2}, respectively.

In variation of parameters, what are the formulas for W, W_{1}, and W_{2}?

What does a CauchyEuler equation look like?
ax^{2 }d^{2}y/dx^{2} + bx dy/dx + cy = 0

What form of equation is ax^{2} d^{2}y/dx^{2} + bx dy/dx + cy = 0?
CauchyEuler

How do you set up a CauchyEuler equation to be solved?
 ax^{2 }d^{2}y/dx^{2} + bx dy/dx + cy = ax^{2}m(m 1)x^{m2} + bxmx^{m1} + cx^{m}
 (Comes from y = cx^{m} and then taking derivatives for a and b)

How do you format the solutions of a CauchyEuler equation?
y = c_{1}x^{m1} + c_{2}x^{m2} or y = c_{1}x^{m1} + c_{2}x^{m1}lnx or y = x^{α}(c_{1}cos β lnx + c_{2}sin β lnx)

What are the equations that would be used to solve a spring problem (4 equations)?
 d^{2}x/dt^{2} + ω^{2}x = 0, ω^{2} = k/m
 x(t) = c_{1}cos ωt + c_{2}sin ωt
 x(t) = Asin(ωt + ϕ), A = √(c_{1}^{2} + c_{22}), tan ϕ = c_{1}/c_{2}
 d^{2}x/dt^{2} + 2λ dx/dt + ω2x = 0

How do you handle the solutions of d^{2}x/dt^{2} + 2λ dx/dt + ω^{2}x = 0?
λ^{2}  ω^{2 }> 0: x(t) = e^{λt}(c_{1e}^{√}

