M427 J Exam 1

int sinx

-cosx

cotx d/dx

-csc^2x

int cscxcotx

-cscx

cscx d/dx

-cscxcotx

int tanx

-ln|cosx| or ln|secx|

int cscx

-ln|cscx+cotx|+c

cosx d/dx

-sinx

linear equations with constant coeffcients

1. characteristic equation 2. find y1 = e^rx and y2 = e^rx ( if r is imaginary then y1 = e^ax cosBx & y2 = e^ax sinBx) if r is repeated y1 = e^rx and y2 = xe^rx 3. gen solution : y(t) = c1y1 +c2y2

classifying method two and three L[y]t = ay" + by' + cy = (...)e^at

1. if a is not a solution to charac. equation { A0 + A1t + A2t^2...+ Ant^n }e^at 2. if a is 1 of 2 solution { A0 + A1t + A2t^2...+ Ant^n }te^at 3. if a is the only solution { A0 + A1t + A2t^2...+ Ant^n }t^2e^at

classifying method one L[y]t = ay" + by' + cy = g(t)

1. if c DOES NOT = 0 then { A0 + A1t + A2t^2...+ Ant^n } 2. if c = 0 and b DOES NOT { A0 + A1t + A2t^2...+ Ant^n }t 3. if they both = 0 { A0 + A1t + A2t^2...+ Ant^n }t^2

Judicious Guessing method 2

L[y]t = ay" + by' + cy = (...)e^at 1. characteristic equation 2. find y1 and y2 and general solution 3. classify v(t) 4. plug into L[v](t) 5. set equal to g(t) 6. solve for An 7. plug An into v(t) 8. y(t) = c1y1 + c2y2 + v(t)

Judicious Guessing method 3

L[y]t = ay" + by' + cy = (...)e^iwt EULERS FORMULA : e^iwt = coswt + isinwt when = to sint then give Im[]t when = to cost then give Re[] t 1. characteristic 2. classify 3. take derivative 4. plug into L[]t 5. set equal to e^iwt 6. solve for A0 7. replace E^iwt in equation 8. find solution 9. can plug back into gen solution too

Judicious Guessing method 1

L[y]t = ay" + by' + cy = g(t) 1. classify v(t) 2. plug into L[v]t= v" + v' + v 3. take derivative 4. set = g(t) 5. solve for An values 6. plug into v(t) equation

sinx d/dx

cosx

population model

dp(t)/dt = ap(t)

hint for nonhomogeneous

dy/dt(e^t) + ty(e^t) = d/dt(e^ty)

L[y]t = 0

general solution : y(t) = c1y1 + c2y2

homogeneous IV

given: y'+a(t)y = 0, y(t0) = y0 y(t)= y0*exp[-int from t0 to t of (a(s)ds)]

L[y](t) = y" + p(t)y' + q(t)y

linear 1. L[cy] = cL[y] 2. L[y1 + y2] = L[y1] + L[y2]

int secx

ln|secx+tanx|

int cotx

ln|sinx|

population model with -bp^2

p(t) = ap0/ (bp0 + (a-bp0)e^(-a(t-t0)))

population model IV

p(t) = p0(-int[a(t)dt]) from t to t0

tanx d/dx

sec^2x

int secxtanx

secx

secx d/dx

secxtanx

int cosx

sinx

L[y]t= y" + p(t)y' + q(t)y = g(t) NON HOMOG.

solution : y(t) = c1y1 + c2y2 + v(t) 1. given solutions 2. find new y1= sol1 - sol2 and y2 = sol3 - sol2 3. plug into general solution

Integration by parts int[udv]

uv-int[u'vdx]

nonhomogeneous linear equation

y' +a(t)y = b(t) 1. m(t) = exp[int(a(t)dt] 2. multiply both sides by m(t). 3. take integral of both sides 4. simplify

homogeneous linear equation

y(t)= exp[-int(a(t)+c1)] y(t)= expc1*exp[-int(a(t))] y(t)= c2exp[-int(a(t))]