# Dif Eq Midterm #2

 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? Cauchy-Euler 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/mx(t) = c1cos ωt + c2sin ωtx(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√ Authorklockhart ID48237 Card SetDif Eq Midterm #2 Descriptiondif eq midterm Updated2010-11-08T23:43:57Z Show Answers