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))]