Dif Eq Midterm #2
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Upon which principle does reduction of order lie?
Given a solution to a differential equation y1, the second solution, y2 can be found by multiplying y1 by a function u(x).
What formula is useful for reduction of order?
- y2(x) = y1(x)∫ e-∫P(x) dx y12 (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 yp if using the method of undetermined coefficients?
Acos(x) + Bsin(x)
What is the basic equation format of variation of parameters?
yp = u1y1 + u2y2
In variation of parameters, how do you find u1 and u2?
u1' = W1/W, u2 = W2/W. Then integrate to find u1 and u2, respectively.
In variation of parameters, what are the formulas for W, W1, and W2?
What does a Cauchy-Euler equation look like?
ax2 d2y/dx2 + bx dy/dx + cy = 0
What form of equation is ax2 d2y/dx2 + bx dy/dx + cy = 0?
How do you set up a Cauchy-Euler equation to be solved?
- ax2 d2y/dx2 + bx dy/dx + cy = ax2m(m -1)xm-2 + bxmxm-1 + cxm
- (Comes from y = cxm and then taking derivatives for a and b)
How do you format the solutions of a Cauchy-Euler equation?
y = c1xm1 + c2xm2 or y = c1xm1 + c2xm1lnx or y = xα(c1cos β lnx + c2sin β lnx)
What are the equations that would be used to solve a spring problem (4 equations)?
- d2x/dt2 + ω2x = 0, ω2 = k/m
- x(t) = c1cos ωt + c2sin ωt
- x(t) = Asin(ωt + ϕ), A = √(c12 + c22), tan ϕ = c1/c2d2x/dt2 + 2λ dx/dt + ω2x = 0
How do you handle the solutions of d2x/dt2 + 2λ dx/dt + ω2x = 0?
λ2 - ω2 > 0: x(t) = e-λt(c1e√
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