Exam 2 Study Guide - Sougata Dhar PDF

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Sougata Dhar...

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Calc 2 Review Recall mathematical notions about power series... 1. If a function f(t) can be written as f(t) = then we say f(t) has a power series representation around to or in the neighborhood of t₀ 2. Every power series is associated with a radius of convergence R > 0, with the following properties A. The power series converges absolutely for R > 0 mean if |t - t₀| < R then the series is convergent. For |t - t₀| > R, the series is divergent. We deal with the end points on a case by case basis. The interval (t₀- R, t₀ + R) is known as interval of convergence B. If R = 0, then the power series converges for t₀ only C. If R = ∞, then the power series converges for all t 3. The ratio/root test are the best way to find the R a. Ratio test: b. Root test: 4. If f(t) has a power series representations at t = t₀ with R > 0, then f(t) is said to be analytics at t = 0. If f(t) is analytic at t₀, then f can be differentiated infinitely many times within the radius of convergence

Some well known Mclaurin Series: Chapter 10 Power Series Solution of 2nd Order Equation Consider x" + p(t)x' + q(t)x = 0 we are looking for x(t) = Σ which solution with | t | < R Definition 1: Ordinary Point Assume x(t) = Σ is a solution of above equation with | t | < R. Suppose to (-R, R). If p(t) and q(t) are both analytic at t₀ then t₀ is an ordinary point of above equation ◦ If either p(t) or q(t) is not continuous at t₀ then t₀ is not an ordinary point Definition 2: Singular Point Recall: x" + p(t)x' + q(t)x = 0 • A point is to be called singular if either p(t) or q(t) fails to be analytic at t₀. Roughly speaking if p(t) or q(t) is discontinuous at t₀ then t₀ is a singular point of the above equation. Consider x"- x' + — x = 0. Here p(t) = -1 and q(t) = — . Clearly q(t) is discontinuous at the origin. Hence t = 0 is a singular point of the equation. Definition 3: Regular singular point A singular point of above equation is said to be regular if the function (t - t₀)p(t) and (t - t₀)²q(t) are analytic at t₀. • If a singular point is not regular then it’s called irregular singular point. Motivation: Assume the equation has a regular singular point t₀ = 0 then there exists a solution in the form x(t) = t Σ =Σ • This is called the Method of Frobenius and the series and x(t) = Σ is called Frobenius series ◦ Includes power series when m = 0 Remark: 1. Once we get one solution using Frobenius method, we can use reduction of order to find the second LI solution 2. One of those two solutions will always have a singularity at t₀ = 0 3. The following is the most general form of a regular singular point at the origin x" + — × Σ p t x' + — × Σ q t x = 0

Chapter 10 Frobenius Method • We use this method to find a power series solution of x" + p(t)x' + q(t)x = 0 near a regular singular point. Remark: Consider x" + p(t)x' + q(t)x = 0 and assume t₀ = 0 is a regular singular point.. i.e. lim tp(t) = p₀ and lim t²q(t) = q₀ . Then x(t) = t Σa t = Σa t , a₀ ≠ 0, is the Frobenius series where m satisfies m(m - 1) + mp₀ + q₀ = 0. This equation is known as the indicial equation . The roots of the indicial equation are called the exponents or characteristic exponents of the above equation.

Summarize Frobenius method Theorem: Consider x" + p(t)x' + q(t)x = 0 suppose t₀ = 0 is a regular singular point and lim tp(t) = p₀ and lim t²q(t) = q₀. Let m₁ and m₂ be the roots of the indicial equation m(m-1) + mp₀ + q₀ = 0 then... 1. If m₁ ≠ m₂ and m₁-m₂ is not an integer, then two LI solutions are given by A. x₁(t) = t Σa t and x₂(t) = t Σa t where a and a can be obtained from the recursive relation 2. if m₁ = m₂ (double root) then x₁ is same as above and x₂(t) = x₁(t)ln|t| + t Σb t , b are determined as before 3. If m₁ ≠ m₂ and m₁-m₂ is an integer then take m₁ be the larger root, I.e. if m₁ and m₂ are complex, take m₁ s.t. m₁ > m₂ here x₁(t) is still same and x₂(t) = C x₁(t) ln |t| + t Σ

Chapter 10 Bessel’s Equation

The Gamma Function: This function is used to give a reasonable meaning to p! or (p + n)! for n {0,1,2,...} and p is a non negative real number. Define Γ(p) = ∫ t ¹e dt p > 0 Note: Γ(p + 1) = pΓ(p) for > 0 Note: In general if p = n we have Γ(n + 1) = n! for nonnegative integers process can be continued indefinitely. • Easy to see that lim Γ(p) = lim —— = ± ∞. In fact this is the way Γ(p) behaves for all negative integers • Can be proved that Γ(½) = √π in fact Γ(p) never vanishes. Hence — is defined. In general Γ(p + 1) = p! for all p except the negative integers Bessel’s DE: The following DE is known as the Bessel’s DE of order m: t²x" + tx' + (t² - m²)x = 0, t = 0 is a singular point (Frobenius solution exists) • x" + —x' + ——x = 0, p(t) = — , q(t) = ——, lim t p(t) = 1 and lim t²q(t) = -m² m is non negative • Bessel’s function of order m of the first kind • Let m = 0, then t²x" + tx' + t²x = 0 or tx" + x' + tx = 0 is called the Bessel’s equation of order 0 • J₀(t) = Σ —— (—) = Σ —— (—) is known as the Bessel’s function of order 0 Bessel’s function of second kind Recall: t = 0 is an RSP of t²x" + tx' + (t² - m²)x = 0 which has the indical equation as p² - m² = 0 - p ± m • If m is not an integer then J (t) and J (t) are two LI solutions and hence x(t) = C₁J (t) + C₂J (t) is the G.S. • When m is an integer a simple calculation shows J (t) = (-1) J (t) I.e. the two functions are L.D. To avoid this we define Y (t) = —————— ◦ This function is known as the Bessel’s function of the second kind. Note, when m is an integer, Y (t) has an inter determinant form. But, the limit always exists... then x(t) = C₁J (t) + C₂Y (t) is the G.S. of Bessel’s DE

Chapter 7 Eigenvalues and first order systems System of first order eq: Suppose A is an n × n matrix. A =

= (a ) moreover, let v be a nonzero vector and λ be a real or complex number. Assume λ and v satisfy the equation Av = λv . Then λ is an eigenvalue of A and v ≠ 0 is the corresponding eigenvectors

• To find λ, we solve det(A - λI) = 0 which is called the “characteristic equation” of A ◦ Above equation yields a polynomial of degree n in λ. This is known as the characteristic polynomial of A Summary: A 2×2 matrix will have 2 eigenvalues. They could be real, same, complex. Each eigenvalue will have a corresponding eigenvector. Any scalar multiple of an eigenvector is also an eigenvector. This result can be extended to a n×n matrix. Theorem: Let A be an m×n matrix with λ 's are distinct real eigenvalues. Assume v₁, ..., v are corresponding. Then v₁, ..., v are linearly independent. First order system We can write a second order equation on a system of first order equations. Discussion- consider x" - 3x' + 2x = 0. This is an easy DE. It should be clear that the auxiliary equation is Chapter 8 Qualitative Analysis of DE Recap: Consider x' = f(t, x): A first order scalar DE t∈R

• Moreover, if x' = a(t)f(x): standard form of separable nonlinear DE • Definition- Autonomous Equation: An autonomous equation is an equation in the standard form where the right hand side of the function is independent of t Lemma-1: Let x(t) be a solution of x' = f(x). Then x(t + n) is a solution of x' = f(x) Plantar Hamiltonian System: We consider a special class of autonomous system as • H(x,y) is a twice differentiable function for (x,y) • This is known as the Hamiltonian system. It’s a conservative system (explained later) Lemma-2: Suppose x(t), y(t) is a solution of HS. Then H(x(t), y(t)) = C for some C in R. Equilibria of HS: The points (x*,y*)∈R s.t. Hₓ(x*, y*) = H (x*, y*) = 0 are called the equilibria of HS Consider the set of curves A = {(x,y)∈R²: H(x,y) = c}

Lemma-3: Assume x (t), y (t) be the solution of HS. Such that H(x(t), y(t)) = C. Set P (t) = (x (t), y (t)). Suppose there exists a T > 0

Exam 2 Solutions 1. Consider the following Cauchy problem

use the eigenvalue method to find the solution

2. Let a be a real number such that a² ≠ 1. Consider the system

Use the Hamiltonian method to find all values of a such that the system has only periodic solutions

3.

The Bessel’s equation of order m ∈ R is given below t²x" + tx' + (t² - m²)x = 0 The Bessel’s function of first kind Jm(t) is a solution of this equation with the form Jm(t) = Σ ——— (—)² Use them to answer the following... (a) Show that J₀ = -J₁ (b) Show that —(tJ₁) = tJ₀ (c) Show that between each pair of consecutive positive zeros of J₀, there is exactly one zero of J₁ and vice versa

4.

Consider the following equation 4t³x" + (cos2t - 1)x' + 2tx = 0 A. Show that t = 0 is a regular singular point B. Find the roots of the indical equation C. Use part B and the Frobenius theorem to write the series form of the two LI solutions D. Find all real numbers a such that the following equation x" + ——x = 0 Does not have a power series solution around the origin. Show complete work and justify conclusion

Chapter 8: Hamiltonian x' = Hy HS = y' = -Hx and H(x,y) = c • Define: {(x,y) ∈ R: H(x,y) = c } which is the solution set of HS • Equilibria of HS: The points (x*, y*)∈ R where Hx(x*, y*) = Hy(x*, y*) = 0 are called the equilibria of HS I.E. set x' = 0 and y' = 0 and solve • Consider {(x,y) ∈ R: H(x,y) = c } (set H(x,y) = 0)...