DET: Module 1&2

Question 1

yc for distinct roots

Answer 1

yc = c1e^m1x + c2e^m2x

Question 2

yc for equal real roots

Answer 2

yc = e^mx(c1 + c2x)

Question 3

yc for complex roots

Answer 3

yc = e^αx[c1cosβx + c2sinβx]

Question 4

yp for ODE

Answer 4

1) Exponentials factor out
2) Polynomials become general degree terms
3) sinx/cosx always appear in pairs

If RHS is (x^2)(e^ax)(sinbx)
yp = e^ax [(Ax^2+Bx+C)sinbx + (Dx^2+Ex+F)cosbx]

for anything else than trig, exp or poly, just make then equal to some variable, say u have lnx, then just put t = lnx

Question 5

Cauchy Euler

Answer 5

1) x^D^2 = D1(D1-1)
xD = D1
2) D1 = d/dz
3) x = e^z

Question 6

Cauchy Legrange

Answer 6

1) (ax+b)^2D^2 = a^2D1(D1-1)
(ax+b)D = aD1
2) D1 = d/dz
3) (ax+b) = e^z

Question 7

LCR circuit eqn

Answer 7

Lq” + Rq’ + q/c = E(t)

Question 8

Damped Oscillations eqn

Answer 8

my” + cy’ + ky = F(t)

1) Free oscillation => F(t) = 0
2) Undamped => cy’ = 0

Question 9

Variation of Parameters

Answer 9

1) Find yc = c1u + c2v
2) let yp = Au + Bv
3) Eqn1 => A’u + B’v = 0
4) Eqn2 => A’u’ + B’v’ = R
5) Solve simultaneously and integrate

Question 10

Soln to PDE: f(p,q) = 0

Answer 10

1) p = a & q=b
2) dz = pdx + qd

Question 11

Soln to PDE: f(p,q,x) = 0

Answer 11

1) q=a
2) dz = pdx + qdy

Question 12

Soln to PDE: f(p,q,y) = 0

Answer 12

1) p=a
2) dz = pdx + qdy

Question 13

Soln to PDE: f(p,q,z) = 0

Answer 13

1) p=aq
2) dz = pdx + qdy

Question 14

Soln to PDE: f(x,p) = f(y,p)

Answer 14

1) f(x,p) = f(y,p) = a
2) dz = pdx + qdy

Question 15

Soln to PDE: z = px + qy + f(p,q)

Answer 15

1) p=a and q=b
2) dz = pdx + qdy

Question 16

Lagrange: Px + Qy = Rz

Answer 16

dx/P = dy/Q = dz/R

Question 17

Separation of Variable

Answer 17

1) let u(x,y) = XY
2) put u=XY in pde
3) get all X in one side and all Y in one side and equate em all to lambda
4) integrate separately