3.1. differential equations

3.1.1. 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: \(r=a\pm bi, y=c_1e^{at} cos(bt)+c_2e^{at} sin(bt)\)

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 \(y_1,y_2\) \(Y_p=-y_1\int \frac{y_2g}{W(y_1,y_2)}dt+y_2\int \frac{y_1g}{W(y_1,y_2)}dt\)

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 \(\implies W=ce^{\int pdt}\)