数学代写|Math220 Differential Equations

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Math220 Differential Equations课程简介

First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: Basic coursework in multivariable calculus and ordinary differential equations, and some prior experience with a proof-based treatment of the material as in Math 171 or Math 61CM.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Prerequisites 

opics covered include: basic methods for solving firstorder and higher-order differential equations with emphasis on linear vs non-linear. Modeling is presented. LaPlace Transforms are developed and used to solve differential and integral equations. Linear systems of differential equations are solved using Eigenvector analysis. Power series as solutions to differential equations are discussed. Optional topics may include, but are not limited to, Fourier Series and Partial Differential Equations.
NOTE:A graphing calculator may be required and will be discussed in class.
Pre-requisite(s):MATH 210 Calculus III.
Terms Offered:Fall, Spring, Summer
Offered Distance Learning:No
Liberal Arts and Sciences Designation: Mathematics
SUNY General Education Designation(s): Mathematics

Math220 Differential Equations HELP（EXAM HELP， ONLINE TUTOR）

To draw a direction field for the differential equation $y^\prime=y$, we need to compute the slope of the tangent line at each point $(t,y)$ in the $ty$-plane. The slope of the tangent line is given by $y^\prime=y$, so it is simply equal to $y$ at each point. We can represent this slope using arrows, pointing in the direction of increasing $y$, as shown below:
Direction Field for $\mathrm{y}^{\prime}=\mathrm{y}$

We can see from the direction field that the solution curves will be exponential functions of the form $y(t) = Ce^t$, where $C$ is a constant determined by the initial condition. As $t$ approaches infinity, the solutions will grow without bound if $C$ is positive, or decay to zero if $C$ is negative.

To show the behavior at $t=\infty$, we can pick a suitable window for our direction field that extends far enough in the $t$-direction to see the asymptotic behavior of the solution curves. For example, we could choose the window $-5 \leq t \leq 5$ and $-5 \leq y \leq 5$, which is large enough to see that the solutions grow or decay rapidly as $t$ moves away from zero. Alternatively, we could choose a narrower window that focuses more closely on the behavior near $t=0$, where the initial conditions have the greatest effect on the solution.

To draw a direction field for the differential equation $y’ = -y$, we first need to find the slope of the tangent line at each point $(t, y(t))$ in the $t-y$ plane. The slope is given by $y'(t) = -y(t)$. We can then plot a small arrow with slope $-y(t)$ at each point $(t, y(t))$.

To choose a suitable window, we need to consider the behavior of the solution at $t=\infty$. The solution to $y’ = -y$ is $y(t) = Ce^{-t}$, where $C$ is a constant. As $t$ increases, the exponential term $e^{-t}$ approaches zero, so the solution approaches zero from either positive or negative direction depending on the sign of $C$. Therefore, we should choose a window that includes $t=0$ and extends far enough in both directions to show the long-term behavior of the solutions.

Here’s a direction field for the differential equation $y’ = -y$ with the window $-5 \leq t \leq 5$ and $-5 \leq y \leq 5$:

The arrows point downward in the upper half-plane ($y > 0$) and upward in the lower half-plane ($y < 0$). The magnitude of the slope increases as we move away from the $t$-axis. As $t$ goes to infinity, the solutions approach zero from either positive or negative direction depending on the sign of the initial condition $y(0)$.

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