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# MATH245 study guide MATH245

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This 15 page Study Guide was uploaded by Jessica Xu on Monday October 3, 2016. The Study Guide belongs to MATH245 at University of Southern California taught by James-Michael Leahy in Fall 2016. Since its upload, it has received 6 views.

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Date Created: 10/03/16

(b) Linear vs. Nonlinear Linear Differential Equation. An nth order ordinary differential equation F(t,y,y,...,y ) = 0 is said to be linear if it can be written in the (n) (n 1) 2 form a (0)y + a1(t)y + ⋯ + a (n)y = g(t). (10) An ODE that is not of the form (10) is a nonlinear equation. The distinction between a linear ODE and a nonlinear ODE hinges only on how the dependent variable y and its derivatives y, y , ... , y (nappear in the equation: for an equation to be linear, they can ap- pear in no other way except as designated by the form (10). Common reasons that an ODE is nonlinear are that there are terms in the equation in which the dependent variable y or any of its derivatives (1) are arguments of a nonlinear function, for example, terms such as −y siny,e ,or (1+y^2)^(1/2) (2) appear as products, or are raised to a 2 ′ power other than 1, such as y and yy. (c) Autonomous vs. Nonautonomous A first order autonomous differential equation is an equation of the form dy/dt = f (y). The distinguishing feature of an autonomous equation is that the independent variable, in this case t, does not appear on the right side of the equation. For instance, the two equations appearing in du/dt =−k(u−T ) and dp/dt =rp−k (1) are autonomous. Other examples of 0 autonomous equations are ′ ′ ′ 2 p = rp(1−p∕K), x = sinx, and y = (k ∕y−1)^(1/2), where r, K, and k are constants. However, the equations u +ku = kT +kAsin���t, x = sin(tx), and ′ 0 y = −y+tare not autonomous because the independent variable t does appear on the right side of each equation. (d) Linear i. Homogeneous vs. Nonhomogeneous Linear Differential Equation. An nth order ordinary differential equation F(t,y,y,...,y ) = 0 is said to be linear if it can be written in the form (n) (n 1) 2 a 0t)y + a 1t)y + ⋯ + a (tny = g(t). (10) The functions a , a ,0...1, a , canled the coefficients of the equation, can depend at most on the independent variable t. Equation (10) is said to be homogeneous if the term g(t) is zero for all t. Otherwise, the equation is nonhomogeneous 3. Qualitative analysis of scalar ODEs: . (a) Equilibrium points . The first step in a qualitative analysis of Eq. dy/dt=f(y)(2) is to find constant solutions of the equation. If y = ���(t) = c is a constant solution of Eq. (2), then dy∕dt = 0. Therefore any constant solution must satisfy the algebraic equation . f (y) = 0. (3) . These solutions are called equilibrium solutions of Eq. (2) because they correspond to no change or variation in the value of y as t increases or decreases. Equilibrium solutions are also referred to as critical points, fixed points, or stationary points of Eq. (2). . . (b) Phase line (use graph of f(y)) . . . . (c) Classify equilibrium points into asympotically stable, unstable, semi stable using phase line and linearization . . (d) Plot direction field with integral curves 4. First order scalar ODEs: . (a) Separable equations . Separable Differential Equation. If the right side f (x, y) of Eq. (3) can be written as the product of a function that depends only on x times another function that depends only on y, dy/dx = f (x, y) = p(x)q(y) . then the equation is called separable . . (b) Solution of linear equation by integrating factor . A differential equation that can be written in the form dy/dt + p(t)y = g(t)is said to be a first order linear equation in the dependent variable y . Use the following steps to solve any first order linear equation. . 1. Put the equation in standard form y + p(t)y = g(t). ′ ∫ p(t) dt . 2. Calculate the integrating factor ���(t) = e . . . (f) Using integrating factors to reduce to an exact equation . . (g) Homogeneous equation . Homogeneous Differential Equation. A differential equation of the form M(x, y)+ N(x,y) dy/dx =0 is homogeneous if M(x,y) and N(x,y) are homogeneous functions of the same degree k. A function f (x, y) is homogeneous of degree k iff (λx, λy) = λ f k (x, y), for all (x, y) in its domain . (h) Bernoulli equation . Bernoulli Differential Equation. A differential equation of the form dy/dt + q(t)y = r(t)y , where n is any real number, is called a Bernoulli equation. . . . 5. 2D Linear Algebra . (a) System of two linear equations . . (b) Determinant and its interpretation . . (c) Inverse of a matrix . (d) Equivalence between invertability and nonzero determinant of a matirx . . . (e) Matrix as a linear transformation on a vector space . . (f) Compute eigenvalues and eigenvectors of a matrix . . . .

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