ODEs

Question 1

list the solns for first order ODEs: homogenous
y’ + p(x) y = 0

Answer 1

y = Ae^-p(x)

Question 2

list the solns for first order ODEs: inhomogenous
y’ + p(x) y = f(x)

Answer 2

y e^∫p(x)dx = ∫ f(x) e^∫p(x) dx
constant is A e^-P(x)
P(x) is integrating factor

or change dy/dx to dx/dy

Question 3

list the solns for first order ODEs: = f(y/x)

Answer 3

u = y/x
y’ = u + x u’
u’ = [f(u) - u] / x

Question 4

list the solns for first order ODEs: y’ + p(x) y = q(x) y^n

Answer 4

z = y^1-n
z’ = (1-n) y^-n y’
z’ + (1-n) p(x) z = (1-n) q(x)

Question 5

second order ODEs: homogenous
y’’ + 2py’ + q(x) = 0

Answer 5

if Y1, Y2 are soln aY1+aY2 are solns

Assume p, q are constant and real
try y=e^λx
λ^2 + 2p(x) λ + q(x) = 0
λ = -p ± √(p^2 - q)

Question 6

homogenous 2nd order 2 real roots

Answer 6

y = Ae^λ1x + Be^λ2x

Question 7

homogenous 2nd order 2 complex

Answer 7

y = e^-px[AcosΩx + BsinΩx]
Re^-px cos(Ωx - ϕ) if A, B = Rcosϕ, Rsinϕ
Ω = √q - p^2

Question 8

homogenous 2nd order p^2 = q

Answer 8

y = (A+Bx)e^-px

Question 9

second order ODEs: explain the physical results for each case (2 real, 1 root, 0 root, yp=yc)
y’’ + 2py’ + q(x) = f(x)

Answer 9

overdamped
critically damped
underdamped
resonance

Question 10

second order ODEs: inhomogeneous method (how to solve yp?)

f is polynomial, exponential, combination of sines and cosines, sum of cases

Answer 10

plug in yp to get constants, then use yp+yc to get boundaries
multiply by x if yp=yc
try y = anx^n + an-1 x^n-1…
y = ae^kx
y = asin(Kx) + bcos(Kx)
find integrals separately and add

Question 11

how to solve higher order ODEs
how to solve (d/dx-2)^4 y=0 (solve it)

Answer 11

try e^λx
remember there will be 4 roots