# 3.1. differential equationsÂ¶

## 3.1.1. Differential EquationsÂ¶

Separable: Separate and Integrate FOLDE: yâ€™ + 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): pyâ€™â€™+qyâ€™+ry=0

Reduction of Order: Know one solution, can find other

Undetermined Coefficients (doesnâ€™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

Abelâ€™s Thm: yâ€™â€™+pyâ€™+q=0 $$\implies W=ce^{\int pdt}$$