gong qingfeng

• Differential Equations

Separable: Separate and Integrate
FOLDE: y’ + p(x)y = g(x)
IF: $e^{int{p(x)}dx}$

Exact: Mdx+Ndy = 0 $M_y=N_x$
Integrate Mdx or Ndy, make sure all terms are present

Constant Coefficients:
Plug in $e^{rt}$, solve characteristic polynomial
repeated root solutions: $e^{rt},re^{rt}$
complex root solutions: $r=apm bi, y=c_1e^{at} cos(bt)+c_2e^{at} sin(bt)$

SOLDE (non-constant):
py’‘+qy’+ry=0

Reduction of Order: Know one solution, can find other

Undetermined Coefficients (doesn’t have to be homogenous): solve homogenous first, then plug in form of solution with variable coefficients, solve polynomial to get the coefficients

Variation of Parameters: start with homogenous solutions $y_1,y_2$
$Y_p=-y_1int frac{y_2g}{W(y_1,y_2)}dt+y_2int frac{y_1g}{W(y_1,y_2)}dt$

Laplace Transforms - for anything, best when g is noncontinuous

$mathcal{L}(f(t))=F(t)=int_0^infty e^{-st}f(t)dt$

Series Solutions: More difficult

Wronskian: $W(y_1 ,y_2)=y_1y _2’ -y_2 y_1’$
W = 0 $implies$ solns linearly dependent

Abel’s Thm: y’‘+py’+q=0 $implies W=ce^{int pdt}$