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Math Tripos Part IA: Differential Equations

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Math Tripos Part IA: Differential Equations Michael Li December 17, 2014 1 Basic calculus Informal treatment of differentiation as a limit, the chain rule, Leibnitz's rule, Taylor series, informal treat- ment of O and o notation and l'H^opitalsrule; integration as an area, fundamental theorem of calculus, integration by substitution and parts. [3] Informal treatment of partial derivatives, geometrical interpretation, statement (only) of symmetry of mixed partial derivatives, chain rule, implicit differentiation. Informal treatment of differentials, including exact differentials. Differentiation of an integral with respect to a parameter. [2] First-order linear differential equations Equations with constant coefficients: exponential growth, comparison with discrete equations, series solution; modelling examples including radioactive decay. Equations with non-constant coefficients: solution by integrating factor. [2] Nonlinear first-order equations Separable equations. Exact equations. Sketching solution trajectories. Equilibrium solutions, stability by perturbation; examples, including logistic equation and chemical kinetics. Discrete equations: equilibrium solutions, stability; examples including the logistic map. [4] Higher-order linear differential equations Complementary function and particular integral, linear independence, Wronskian (for second-order equa- tions), Abel's theorem. Equations with constant coefficients and examples including radioactive sequences, comparison in simple cases with difference equations, reduction of order, resonance, transients, damping. Homogeneous equations. Response to step and impulse function inputs; introduction to the notions of the Heaviside step-function and the Dirac delta-function. Series solutions including statement only of the need for the logarithmic solution. [8] Multivariate functions: applications Directional derivatives and the gradient vector. Statement of Taylor series for functions on Rn. Local extrema of real functions, classification using the Hessian matrix. Coupled first order systems: equivalence to single higher order equations; solution by matrix methods. Non-degenerate phase portraits local to equilibrium points; stability. Simple examples of first- and second-order partial differential equations, solution of the wave equation in the form f(x + ct) + g(x − ct). [5] 2 Contents 1 Differentiation 4 1.1 Differentiation . 4 1.2 Small o and big O notations . 4 1.3 Methods of Differentiation . 4 1.4 L'Hopital's Rule . 5 2 Integration 5 2.1 Integration . 5 2.2 Methods of integration . 6 2.3 Integration by Parts . 6 3 Partial Differentiation 6 3.1 partial Differentiation . 6 3.2 Chain rule . 7 3.3 Implicit Differentiation . 7 3.4 Differentiation of Integral w.r.t parameter in Integrand . 8 4 First-order differential equation 8 4.1 The Exponential Function . 8 4.2 First-order Linear Homogenous Ordinary Differential Equation . 9 4.2.1 Discrete equation . 9 4.3 Forced (inhomogeneous) equations . 10 4.4 Non-constant coefficients . 10 4.5 Nonlinear first-order equations . 10 4.5.1 Separable Equations . 10 4.5.2 Exact Equations . 11 4.6 Solution curves (Trajectories) . 12 4.6.1 Stability of Fixed Points . 12 4.6.2 Perturbation Analysis . 12 4.6.3 Autonomous systems . 13 4.7 Logistic Equation . 13 4.7.1 Fighting for limited resources . 13 4.8 Discrete equations(Difference Equation) . 14 4.8.1 Behaviour of Logistic map . 14 5 Second-order Differential Equations 14 5.1 Constant Coefficients . 14 5.1.1 Complementary function . 15 5.1.2 Detuning . 15 5.1.3 Method for Second Complementary Function . 15 5.1.4 Phase Space . 16 5.2 Particular Integrals . 16 5.2.1 Guessing . 16 5.2.2 Detuning . 17 3 5.2.3 Variation of parameters . 17 5.3 Homogeneous equations, a.k.a Linear equidimensional equations . 17 5.3.1 Solving . 17 5.4 Difference Equation . 18 5.4.1 Guesswork . 18 5.5 Transcients and damping . 18 5.5.1 Free (natural) response f =0........................... 18 5.5.2 Underdamped . 19 5.5.3 Critically damped . 19 5.5.4 Overdamped . 19 5.5.5 Forcing . 19 5.6 Impulses and Point Forces . 19 5.7 Heaviside Step Function . 20 5.7.1 Solving differential equations with discontinuous functions . 20 6 Series Solution 20 6.1 Behaviour near x = x0 ................................... 22 7 Directional Derivatives 23 7.0.1 The gradient vector of f(x; y)........................... 23 7.1 Stationary Points . 23 7.2 Taylor series for Multivariable Functions . 23 7.3 Classification of stationary points . 24 7.3.1 Determination of definiteness . 24 7.3.2 Contours of f(x; y)................................. 25 8 Systems of Linear differential equations 25 8.1 Phase-space trajectories . 25 8.2 Nonlinear dynamical systems . 26 8.2.1 Equilibrium (fixed) points . 27 8.2.2 Stability . 27 9 Partial differential equations (PDEs) 27 9.1 First-order wave equation . 27 9.2 Second-order wave equation . 28 10 The Diffusion Equation 29 4 1 Differentiation 1.1 Differentiation Definition. The derivative of a function fx w.r.t to an independent variable x is defined as the rate of change of f(x) with x, or symbolically: df f(x + h) − f(x) = lim dx h!0 h f(x) is differentiable at a point x = a iff the limit exists. (jxj is not differentiable at x = 0) df 0 d d2f Note. dx ; f (x); dx f(x) all mean the same thing. Higher derivatives follow the same rule ( dx2 = 00 d d 0 f (x) = dx ( dx f(x))). However, f represents the derivative w.r.t the argument. 1.2 Small o and big O notations Definition. f(x) (i) f(x) = o(g(x)) as x ! x0 if lim g(x) = 0, so f(x) is much smaller than g(x). x!x0 f(x) (ii) f(x) = O(g(x)) as x ! x0 if g(x) is bounded as x ! x0, so f(x) is about the size of g(x). Note. Bounded doesn't mean limit exists. f(x) = o(g(x)) ) f(x) = O(g(x)) 0 Corollary. f(x0 + h) = f(x0) + f (x0)h + o(h) Proof. f(x + h) − f(x ) 0(h) f 0(x ) = 0 0 + 0 h h from definition of the derivative and the small o notation. Result follows. 1.3 Methods of Differentiation Theorem (Chain rule). Given f(x) = F (g(x)), then df dF dg = : dx dg dx dg Proof. Assuming that dx exists and is therefore finite, we have df F (g(x + h)) − F (g(x)) = lim dx h!0 h F (g(x)) + (hg0(x) + o(h))F 0(g(x)) + o(hg0(x) + o(h)) − F (g(x)) = lim h!0 h df dg = g0(x)F 0(g(x)) = dg dx 5 Theorem (Leibinz's Rule). Given f = uv, then n X n f (n)(x) = u(r)v(n−r) r r=0 where f (n) is the n-th derivative of f. 1.4 L'Hopital's Rule Theorem. If f(x); g(x) differentiable at x0, and lim f(x) = lim g(x) = 0, then: x!x0 x!x0 f(x) f 0(x) lim = lim 0 : x!x0 g(x) x!x0 g (x) 0 Proof. From the Taylor's Theorem, we have f(x) = f(x0)+(x−x0)f (x0)+o(x−x0), and similarly for g(x). Thus 0 f(x) f(x0) + (x − x0)f (x0) + o(x − x0) lim = lim 0 x!x0 g(x) x!x0 g(x0) + (x − x0)g (x0) + o(x − x0) 0 o(x−x0) f (x0) + = lim x−x0 o(x−x ) x!x0 g0(x ) + 0 0 x−x0 f 0(x) = lim 0 x!x0 g (x) 2 Integration 2.1 Integration Definition. An integral is a limit of an (infinite) sum, or the "area under the graph": N Z b X f(x) dx = lim f(xn)∆x: ∆x!0 a n=0 If f is differentiable, the total area under the graph from a to b is: N−1 N−1 Z b X 2 X lim (f(xn)∆x) + N · O(∆x ) = lim (f(xn)∆x) + O(∆x) = f(x) dx N!1 N!1 n=0 n=0 a due to the missing "triangle"O(∆x2) of the area in the sum. R x 0 Theorem (Fundamental Theorem of Calculus). Let F (x) = a f(t) dt. Then F (x) = f(x). 6 Proof. d 1 Z x+h F (x) = lim f(t) dt dx h!0 h x 1 = lim [f(x)h + O(h2)] h!0 h = f(x) d R b d R g(x) 0 Similarly, we have dx x f(t) dt = −f(x) and dx a f(t) dt = f(g(x))g (x). Note. We write R f(x) dx = R x f(t) dt,; the unspecified lower limit gives constant +C 2.2 Methods of integration Useful Hints for trig substitution: Useful identity pPart of integrand Substitution cosθ + sinθ = 1 1 − x2 x = sin θ 2 2 2 1 + tan θ = sec θ p1 + x x = tan θ 2 2 2 cosh u − sinh u = 1 px − 1 x = cosh u cosh2 u − sinh2 u = 1 1 − x2 x = sinh u 1 − tanh2 u = sech2u 1 − x2 x = tanh u 2.3 Integration by Parts Integrating the product rule, we have: Z Z uv0 dx = uv − vu0 dx: Example 1 (THE example for Integration by Parts). Consider R log x dx. Let u = log x and 0 0 1 v = 1. Then u = x and v = x. So we have Z Z log x dx = x log x − dx = x log x − x + C 3 Partial Differentiation 3.1 partial Differentiation Definition (Partial derivative). Given a function of several variables f(x; y), the partial derivative of f with respect to x is the rate of change of f as x varies, keeping y constant.
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