ODE Cheat Sheet, Cheat Sheet of Mathematics

Download ODE Cheat Sheet and more Cheat Sheet Mathematics in PDF only on Docsity! ODE Cheat Sheet First Order Equations Separable y′(x) = f(x)g(y)∫ dy g(y) = ∫ f(x) dx+C Linear First Order y′(x)+p(x)y(x) = f(x) µ(x) = exp ∫ x p(ξ) dξ Integrating factor. (µy)′ = fµ Exact Derivative. Solution: y(x) = 1 µ(x) (∫ f(ξ)µ(ξ) dξ + C ) Exact 0 = M(x, y) dx+N(x, y) dy Solution: u(x, y) = const where du = ∂u ∂x dx+ ∂u ∂y dy ∂u ∂x = M(x, y), ∂u ∂y = N(x, y) Condition: My = Nx Non-Exact Form µ(x, y) (M(x, y) dx+N(x, y) dy) = du(x, y) My = Nx N ∂µ ∂x −M ∂µ ∂y = µ ( ∂M ∂y − ∂N ∂x ) . Special cases If My−Nx M = h(y), then µ(y) = exp ∫ h(y) dy If My−Nx N = −h(x), then µ(y) = exp ∫ h(x) dx Second Order Equations Linear a(x)y′′(x) + b(x)y′(x) + c(x)y(x) = f(x) y(x) = yh(x) + yp(x) yh(x) = c1y1(x) + c2y2(x) Constant Coefficients ay′′(x) + by′(x) + cy(x) = f(x) y(x) = erx ⇒ ar2 + br + c = 0 Cases Distinct, real roots: r = r1,2, yh(x) = c1e r1x + c2er2x One real root: yh(x) = (c1 + c2x)e rx Complex roots: r = α± iβ, yh(x) = (c1 cosβx+ c2 sinβx)eαx Cauchy-Euler Equations ax2y′′(x) + bxy′(x) + cy(x) = f(x) y(x) = xr ⇒ ar(r − 1) + br + c = 0 Cases Distinct, real roots: r = r1,2, yh(x) = c1x r1 + c2xr2 One real root: yh(x) = (c1 + c2 ln |x|)xr Complex roots: r = α± iβ, yh(x) = (c1 cos(β ln |x|) + c2 sin(β ln |x|))xα Nonhomogeneous Problems Method of Undetermined Coefficients f(x) yp(x) anxn + · · ·+ a1x+ a0 Anxn + · · ·+A1x+A0 aebx Aebx a cosωx+ b sinωx A cosωx+B sinωx Modified Method of Undetermined Coefficients: if any term in the guess yp(x) is a solution of the homogeneous equation, then multiply the guess by xk, where k is the smallest positive integer such that no term in xkyp(x) is a solution of the homogeneous problem. Reduction of Order Homogeneous Case Given y1(x) satisfies L[y] = 0, find second linearly independent solution as v(x) = v(x)y1(x). z = v′ satisfies a separable ODE. Nonhomogeneous Case Given y1(x) satisfies L[y] = 0, find solution of L[y] = f as v(x) = v(x)y1(x). z = v′ satisfies a first order linear ODE. Method of Variation of Parameters yp(x) = c1(x)y1(x) + c2(x)y2(x) c′1(x)y1(x) + c ′ 2(x)y2(x) = 0 c′1(x)y ′ 1(x) + c ′ 2(x)y ′ 2(x) = f(x) a(x) Applications Free Fall x′′(t) = −g v′(t) = −g + f(v) Population Dynamics P ′(t) = kP (t) P ′(t) = kP (t)− bP 2(t) Newton’s Law of Cooling T ′(t) = −k(T (t)− Ta) Oscillations mx′′(t) + kx(t) = 0 mx′′(t) + bx′(t) + kx(t) = 0 mx′′(t) + bx′(t) + kx(t) = F (t) Types of Damped Oscillation Overdamped, b2 > 4mk Critically Damped, b2 = 4mk Underdamped, b2 < 4mk Numerical Methods Euler’s Method y0 = y(x0), yn = yn−1 + ∆xf(xn−1, yn−1), n = 1, . . . , N. Series Solutions Taylor Method f(x) ∼ ∑∞ n=0 cnxn, cn = f(n)(0) n! 1. Differentiate DE repeatedly. 2. Apply initial conditions. 3. Find Taylor coefficients. 4. Insert coefficients into series form for y(x). Power Series Solution 1. Let y(x) = ∑∞ n=0 cn(x− a)n. 2. Find y′(x), y′′(x). 3. Insert expansions in DE. 4. Collect like terms using reindexing. 5. Find recurrence relation. 6. Solve for coefficients and insert in y(x) series. Ordinary and Singular Points y′′ + a(x)y′ + b(x)y = 0. x0 is a Ordinary point: a(x), b(x) real analytic in |x− x0| < R Regular singular point: (x− x0)a(x), (x− x0)2b(x) have convergent Taylor series about x = x0. Irregular singular point: Not ordinary or regular singular point. Frobenius Method 1. Let y(x) = ∑∞ n=0 cn(x− x0)n+r. 2. Obtain indicial equation r(r − 1) + a0r + b0. 3. Find recurrence relation based on types of roots of indicial equation. 4. Solve for coefficients and insert in y(x) series. Laplace Transforms Transform Pairs c c s eat 1 s− a , s > a tn n! sn+1 , s > 0 sinωt ω s2 + ω2 cosωt s s2 + ω2 sinh at a s2 − a2 cosh at s s2 − a2 H(t− a) e−as s , s > 0 δ(t− a) e−as, a ≥ 0, s > 0