Differential equations

Mathematics  ·  chapter 4 of 8  ·  1 min read

Differential equations#

A cheat sheet of methods for solving ordinary differential equations.

About this chapter

A cheat sheet of solution methods, one or two lines each. First-order equations come first: separable ones, first-order linear ones (FOLDE, solved with the integrating factor \(e^{\int p(x)dx}\)) and exact ones.

The rest is about second-order linear equations. With constant coefficients you plug in \(e^{rt}\) and solve a polynomial for \(r\). Undetermined coefficients and variation of parameters handle a forcing term \(g\), the Wronskian tests whether two solutions are independent, and Laplace transforms are the tool of choice when \(g\) is discontinuous.

Differential Equations#

Each line is a method. First-order equations are separable, linear (solved with an integrating factor) or exact. Second-order linear equations with constant coefficients reduce to the roots of a polynomial; a forcing term \(g\) is handled by undetermined coefficients, variation of parameters or Laplace transforms.

Interactive 1. The constant-coefficient case \(y'' + py' + qy = 0\), starting from \(y(0) = 1\), \(y'(0) = 0\). Plugging in \(e^{rt}\) gives \(r^2 + pr + q = 0\). Raise \(p\) and the complex pair \(r = a \pm bi\) slides left and the oscillation \(e^{at}\cos(bt)\) dies faster; at \(p^2 = 4q\) the roots meet, and past that they are real and the solution stops oscillating. Make \(p\) negative to move the roots into the right half-plane and the solution grows.

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

Key ideas

  • A separable equation is solved by moving each variable to its own side and integrating.

  • A first-order linear equation \(y' + p(x)y = g(x)\) is solved by multiplying through by the integrating factor \(e^{\int p(x)dx}\).

  • With constant coefficients, plugging in \(e^{rt}\) turns the equation into a polynomial, and complex roots \(a \pm bi\) give oscillations \(e^{at}\cos(bt)\) and \(e^{at}\sin(bt)\).

  • With a forcing term, solve the homogeneous equation first, then find a particular solution by undetermined coefficients or variation of parameters.

  • Two solutions are independent when their Wronskian \(W = y_1 y_2' - y_2 y_1'\) is nonzero.

  • Laplace transforms work for any forcing term and are the best choice when \(g\) is discontinuous.

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