Once More, From Memory
Jun. 5th, 2006 01:05 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
quasigeostrophy2nd DiffEQs test tonight.
4.1: Linear Independence & Wronskian:
W(y1,y2)=y1*y2'-y2*y1', if W <> 0, y1 & y2 are linearly independent & form a fundamental solution set.
4.2: 2nd Order Linear Homogeneous DEs w/ Constant Coefficients:
Factor the characteristic equation.
1) m has distinct, real roots: y=c1e^(m1x)+c2e^(m2x)
2) m has repeated roots of multiplicity k: y=c1e^(mx)+c2xe^(mx)+...+ckx^(k-1)e^(mx)
3) m has complex conjugate roots: y=e^(ax)(c1cos(bx)+c2sin(bx))
4.3: Reduction of Order:
y2=uy1
w=u'
4.4: Undetermined coefficients - superposition approach.
yp=G(x) where G(x) has like terms to g(x), multiplied through by high enough powers of x to eliminate duplicate terms with yc
4.5: Differential Operators.
Express D in terms to annihilate f(x).
4.7: Variation of Parameters:
yp=u1y1+u2y2 where
u1'=W1/W, u2'=W2/W
W1=-y2f, W2=y1f
5.1: Simple Harmonic Motion:
mx''+kx=0, w^2=k/m
x=c1cos(wt)+c2sin(wt)=Asin(wt+f), A=sqrt(c1^2+c2^2), tanf=c1/c2
5.2: Damped Harmonic Motion:
mx''+Bx'+kx=0, B=2L
1) L > w (overdamped): x=c1e^(-(L-sqrt(L^2-w^2))t)+c2e^(-(L+sqrt(L^2-w^2))t)
2) L = w (critically damped): x=c1e^(-Lt)+c2te^(-Lt)
3) L < w (underdamped): x=e^(-Lt)(c1cos(sqrt(w^2-L^2)t)+c2sin(sqrt(w^2-L^2)t)
5.3: Forced Oscillation:
(as above)=f(t) (non-homogenous)
6.1: Cauchy-Euler Equations:
y=x^m
1) distinct roots: y=c1x^m1+c2x^m2
2) repeated roots multiplicity k: y=c1x^m+c2x^m(lnx)+ckx^m(lnx)^k-1
3) complex conjugate roots: y=x^a(c1cos(Blnx)+c2sin(Blnx))
reduction to constant coefficients: x=e^t, t=lnx, dt/dx=1/x=e^-t
y'=dy/dx=dy/dt*dt/dx=e^-tdy/dt
y''=e^(-2t)d2y/dt2-e^(-2t)dy/dt
The procedures for this test seem much easier than for the first one, but this test also involves some memorization, which I despise. Not that I can't do it, but memorization in many circumstances seems like wasted effort. I'm much better at remembering algorithms/procedures. Going to go through each section and do one more problem from each before I head down to campus.
4.1: Linear Independence & Wronskian:
W(y1,y2)=y1*y2'-y2*y1', if W <> 0, y1 & y2 are linearly independent & form a fundamental solution set.
4.2: 2nd Order Linear Homogeneous DEs w/ Constant Coefficients:
Factor the characteristic equation.
1) m has distinct, real roots: y=c1e^(m1x)+c2e^(m2x)
2) m has repeated roots of multiplicity k: y=c1e^(mx)+c2xe^(mx)+...+ckx^(k-1)e^(mx)
3) m has complex conjugate roots: y=e^(ax)(c1cos(bx)+c2sin(bx))
4.3: Reduction of Order:
y2=uy1
w=u'
4.4: Undetermined coefficients - superposition approach.
yp=G(x) where G(x) has like terms to g(x), multiplied through by high enough powers of x to eliminate duplicate terms with yc
4.5: Differential Operators.
Express D in terms to annihilate f(x).
4.7: Variation of Parameters:
yp=u1y1+u2y2 where
u1'=W1/W, u2'=W2/W
W1=-y2f, W2=y1f
5.1: Simple Harmonic Motion:
mx''+kx=0, w^2=k/m
x=c1cos(wt)+c2sin(wt)=Asin(wt+f), A=sqrt(c1^2+c2^2), tanf=c1/c2
5.2: Damped Harmonic Motion:
mx''+Bx'+kx=0, B=2L
1) L > w (overdamped): x=c1e^(-(L-sqrt(L^2-w^2))t)+c2e^(-(L+sqrt(L^2-w^2))t)
2) L = w (critically damped): x=c1e^(-Lt)+c2te^(-Lt)
3) L < w (underdamped): x=e^(-Lt)(c1cos(sqrt(w^2-L^2)t)+c2sin(sqrt(w^2-L^2)t)
5.3: Forced Oscillation:
(as above)=f(t) (non-homogenous)
6.1: Cauchy-Euler Equations:
y=x^m
1) distinct roots: y=c1x^m1+c2x^m2
2) repeated roots multiplicity k: y=c1x^m+c2x^m(lnx)+ckx^m(lnx)^k-1
3) complex conjugate roots: y=x^a(c1cos(Blnx)+c2sin(Blnx))
reduction to constant coefficients: x=e^t, t=lnx, dt/dx=1/x=e^-t
y'=dy/dx=dy/dt*dt/dx=e^-tdy/dt
y''=e^(-2t)d2y/dt2-e^(-2t)dy/dt
The procedures for this test seem much easier than for the first one, but this test also involves some memorization, which I despise. Not that I can't do it, but memorization in many circumstances seems like wasted effort. I'm much better at remembering algorithms/procedures. Going to go through each section and do one more problem from each before I head down to campus.
