Thing to Definitely Know Euler Identity Pythagorean Identity Trigonometric Identitie e iθ co θ + i in θ co 2 θ + in 2 θ I Firt Order Differential Equation co(u + v co u co v in u in v in(u + v co u in v + in u co v co 2 u ( + co 2u 2. Linear Equation y + py g. Multiply by the integrating factor µ e p : y(t ( t µ(tg(t dt + c µ(t t Compare with Variation of Parameter below. Velocity 2. Separable Equation: y f(xg(y. Seperate and integrate ide eparately: g(y dy f(x dx + c Solve for y when poible. Example: Tank Mixing, Continuouly Compounded Interet, 3. Exact Equation: M(x, y dx + N(x, y dy. The equation i an exact differential form if dψ ψ Check exactne by checking that M ψ(x, y k. dx + ψ dy M(x, y dx + N(x, y dy N. If exact then the olution are the level et of a potential function ψ(x, y ( M + N M Example: Population Dynamic, Black Hole Evaporation, Newton Law of Cooling II Euler Method A olution to an initial value problem y(t y and y f(y, t can be etimated numerically by a piecewie linear function. 2 Euler method with tep ize h etimate a olution iteratively by etting t n t n + hy n y n + hf(y n, t n 3 The Mean Value Theorem guarantee there i ome unknown time a (t n, t n+ uch that y(t n+ y(t n + hy (t n + h2 2 y (a 4 The local truncation error in Euler method can be bounded by y n y(t n h2 2 max y 5 The total local truncation error in Euler method from t to t n can be etimated a h 2 (t n t max y
III Nonlinear Autonomou Sytem of Differential Equation: ( ( d x f(x, y dt y g(x, y ( where x and y are function of time t and f and g are function of x and y only. More generally ẋ F (x. Often not olveable analytically. Equilibrium point or critical point or tationary point are point where the derivative F (x vanihe. In a 2 dimenional ytem, the equilibrium (x, y atifie ( ( ( d x f(x, y dt y g(x, y xx,yy 2 Almot Linear Sytem A nonlinear equation may be approximated by a linear equation near equilibrium point a (or (x, y uing the Taylor expanion d dt x F (a(x a for x near a, where F i the Jacobian derivative. For a 2 dimenional ytem for (x, y near (x, y ( d x dt y ( f f xx,yy ( x x y y 3 Stability Analyi The Jacobian derivative at the equilibrium point give tability: J ( f Equilibrium (x, y i attracting if every eigenvalue of J(x, y i negative, repelling if every eigenvalue i poitive, and a addle if there are eigenvalue of mixed ign. 4 Solution trajectorie are the curve in phae pace (or x-y pace traced by olution to equation (. Trajectorie are olution to the (ometime olveable differential equation f dy dx dy/dt f(x, y dx/dt g(x, y 5 A differential equation or in general any dynamical ytem i chaotic if it i enitive to initial condition, 2 the time evolution of any two region eventually overlap, and 3 every point i arbitrarily cloe to a periodic orbit. 6 Nonlinear differential equation might have trange chaotic attractor where olution are chaotic. Strange attractor may be complicated et, but nearby olution will move toward the attractor. 7 Individual numerical olution to chaotic equation are unreliable, but the location and hape of trange attractor can be etimated numerically. IV Homogenou Linear Sytem: where A i a n n matrix valued function and x i a vector valued fucntion. x (t A(tx(t (2 If x(t and y(t are olution to equation (2 then o i any linear combination ax(t + by(t. 2 Equation (2 ha n linearly independent olution away from the dicontinuitie of A. 3 Solution x,..., x n are linearly independent in interval [a, b] if and only if the Wronkian i non-zero in [a, b] W (t det[x (t... x n (t] 4 If x,..., x n are linearly independent olution then the matrix with x,..., x n a column vector i a fundamental matrix. χ(t [x (t... x n (t] 5 Subject to the initial condition x(t x with t in interval [a, b] with det χ(t for every t [a, b] then there i a unique olution x(t χ(tχ (t x
6 One can tranform an n th order linear differential eqation into a linear ytem: y (n a (ty(t + a (ty... + a n (ty (n become d dt y y. y (n. I n a (t... a n (t................ a (t a (t... a n 2 (t a n (t V Autonomou Homogenou Linear Sytem: where A R n n i a contant matrix and x i a vector valued function. Eigenvalue λ are root of the characteritic polynomial Eigenvalue are real or come in complex conjugate pair. x (t Ax(t (3 p(λ det(a λi 2 The algebraic multiplicity of the eigenvalue i it multiplicity a a root of the characteritic polynomial. 3 Eigenvector are linearly independent olution to 4 If v i an eigenvector aociated to eigenvalue λ then i a olution to Equation (3. (A λiv x(t e λt v 5 If v ± u ± iw are a pair of complex eigenvector correpponding to conjugate pair of eigenvalue λ ± α ± iβ then the real part of the pan of the olution e λ+t v + and e λ t v i the pan of olution e α (co βtu in βtw and e α (in βtu + co βtw 6 The rank of (A λi i the number of linearly independent column. rank(a λi number of linearly independent row number of pivot in the reduced row echelon form dimenion of the range 7 The geometric multiplicity of λ i the number of linearly independent eigenvector of A aociated to λ. By the rank-nullity theorem, the geometric multiplicity of λ i n rank(a λi 8 Algebraic multiplicity geometric multiplicity. If algebraic multiplicity of λ > geometric multiplicity of λ, then λ i a defective eigenvalue and A i a defective matrix. 9 A generalized eigenvector i a vector w uch that (A λi k w for ome k. The mallet uch k i the rank of the generalized eigenvector w. For every n n matrix A there i a bai of bai of generalized eigenvector V [v... v n ] that block-diagonalize A into Jordan Canonical form. A V JV where J i a block diagonal matrix with Jordan block of the form λ λ........... λ λ The column of uch a change of bai matrix V may be computed by forming Jordan chain w n A λi w n A λi... A λi w A λi of generalized vector by iteratively chooing olution to (A λiw k+ w k for k... n where w. The length of all Jordan chain for eigenvalue λ um to the algebraic multiplicity m. The number of Jordan chain for eigenvalue λ i the geometric multiplicity.
A λi A λi 2 If w n w n... A λi A λi w i a Jordan chain then there i a linearly independent olution to ẋ Ax for every generalized eigenvector in the chain given by ( e λt w k + tw k +... + tk (k! w for each k... n. 3 Equivalently, if λ ha algebraic multiplicity m and v,..., v m pan the nullpace of (A λi m, then for k,..., m give m linearly independent olution to equation (3. x k (t tm e λt m! (A λim v k +... + e λt v k 4 If χ(t i a fundamental matrix then the matrix exponential i the unique fundemental matrix normalized at e At n t n n! An χ(tχ ( 5 The general olution to equation (3 i x(t e At c where c R n i a contant vector. 6 A i invertible if and only if it ha only non-zero eigenvalue if and only if det A. 7 If A i invertible than the dynamical ytem d dtx Ax ha exactly one equilibrium point: The origin. The equilibrium point i table or attracting if all the eigenvalue of A have negative real part, untable or repelling if all the eigenvalue have poitive real part, and a addle of emitable if the eigenvalue have mixed ign. If the eigenvalue are complex then olution piral or ocillate. If the eigenvalue are completely imaginary, then the equilibrium point i a piral center. 8 The phae portrait of the dynamical ytem d dt x Ax i a viual dicription of olution trajectorie in Rn and enable a graphical analyi of long term olution behavior ( t. Solution trajectorie hould demontrate the eigenline, the dominant olution behavior, and the direction in which all trajectorie are followed. Direction field are alo a plu. 9 To find the invere of A rref[ A I ] [ I A ] 2 If A i invertible then d dt x Ax+b can be olved by a change of variable to y x+a b and olving d dt y Ay. VI Nonhomogenou Linear Sytem: x (t P(tx(t + g(t (4. Find a fundemental et of olution x,..., x n to the homogenou equation x (t P(tx(t. Let χ [x... x n ] be the correponding fundamental matrix. 2. A particular olution to the nonhomogenou equation i given by x p (t χ χ g(t dt 3. The general olution to the nonhomogenou equation (4 i then x(t c x (t +... + c n x n (t + x p (t or VII Second Order Linear Equation: ( x(t χc + x p χ χ g dt + c y + p(ty + q(ty g(t (5. Given two olution y (t and y 2 (t the Wronkian i ( y (t y W (t det 2 (t y (t y 2(t The Wronkian i nonzero wherever the olution are linearly indepedent.
2. The general olution to equation (5 i y(t c y (t + c 2 y 2 (t + y p (t where y and y 2 are linearly indepedent olution to the homogenou equation and y p i a particular olution to equation (5. y + p(ty + q(ty 3. Variation of Parameter: If y and y 2 are homogenou olution to equation (5 with Wronkian W then a particular olution i y2 (tg(t y (tg(t y p (t y (t dt + y 2 (t dt W (t W (t VIII Contant Coefficient Second Order Linear Equation: my + by + ky g(t (6. If m, b, k > the equation can be interpreted a the Newtonian equation of motion for a ma pring ytem of ma m, damping contant b, and Hooke contant k under a driving force of g(t. 2. The characteritic polynomial of equation (6 i p(λ mλ 2 + bλ + k. Homogenou olution depend on the root of the characteritic polynomial. If λ, λ 2 are the root of p: Root Homogenou Solution Dicriminant Spring Cae λ λ 2 R e λt e λ2t b 2 < 4mk Overdamped λ λ 2 R e λt te λ2t b 2 4mk Critically damped λ ± α ± iω e αt in ωt e αt co ωt b 2 > 4mk Underdamped λ ± ±iω in ωt co ωt b Undamped 3. Method of Undetermined Coefficient: Determine a particular homogenou olution by plugging the anzatz into the differential equation and attempting to fix the unknown contant. In general, if the inhomogeneity i of the form p(te λt for a polynomial p, then you hould gue t m q(te λt where m i the algebraic multiplicity of λ a an eigenvalue and q i a polynomial with unknown coefficient and deg q deg p. Similarly Inhomogeneity: p(te αt in ωt p(te αt co ωt Anatz: t m e αt q(t in ωt + t m e αt q(t co ωt where m i the algebraic multiplicity of α ± ω and q, q are degree deg p polynomial of with unknown coefficient. IX Laplace Tranform: The Laplace Tranform i a linear operator which act on a function by f L[f](. Linear: L [af(t + bg(t] ( al[f(t]( + bl[g(t]( 2. Invertible: there i an invere linear tranform L uch that e t f(t dt L [L [f(t]] f(t for any piecewie continuou, exponentially dominated function f : [, R. 3. Derivative tranform to multiplication by the frequency: [ ] d L dt f(t ( L [f(t] ( f(
4. Exponential in time tranform to hift in frequency: 5. Multiplication by time tranform to derivative: L [ e at f(t ] ( L [f(t] ( a L [tf(t] ( d L [f(t] ( d 6. A piecewie continuou, exponentially dominated function atifie: 7. Time dilation give invere frequency dilation lim L [f(t] ( L [f(at] ( ( a L [f(t] a 8. To olve a differential equation in independent variable y: Tranform an differential equation in time t to an algebraic equation in term of the Laplace variable, then olve for L [y] in term of and invert the tranform. Invert by mean of a Laplace tranform table (learn to ue the table on page 328 and the method of partial fraction decompoition. 9. Write piecewie continuou function uing the unit tep or Heaviide function { if t c u c (t u(t c if t < c which atifie L [u(t cf(t c] ( e c L [f(t] (. The impule, point ma, or δ-dirac function δ(t may be thought of a the (ditributional derivative of the Heaviide function. It ha the property: b a δ(t t f(tdt f(t if t (a, b and if t / [a, b]. L [δ(t].. A periodic function f with period T atifie f(t + T f(t for any argument value t. If f i periodic then L [f] ( T f(te t dt e T L [f(t ( u(t T ] ( e T 2. The convolution of a function f and g i written f g and defined by f g(t t f(t ug(u du The convolution i commutative f g g f and linear in each argument. 3. Convolution tranform to multiplication L [f g] ( L [f] (L [g] ( Laplace Tranform L Time Frequency Time Frequency f(t L [F ](t L[f]( F ( L [F ](t f(t F ( L[f]( f + g L[f] + L[g] cf cl[f] f L[f] f( f (n n F ( n f(... f (n ( tf(t d d F ( tn f(t ( n F (n ( T f(te t dt f(t f(t + T f g(t e T t f(t τg(τdτ L [f] L [g] f(at a F ( a a f ( t a F (a δ(t c e c e λt λ e λt f(t F ( λ t n n! t p Γ(p+ for p > n+ p+ ω in ωt 2 +ω co ωt 2 2 +ω 2 inh at 2 a coh at 2 2 a 2 e u(t c c u(t cf(t c e c L[f(t]( Heaviide unit tep function u, Dirac delta δ, gamma function Γ