Differential Equation Tutorial 1
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This tutorial document provides 4 partial differential equations and asks the reader to determine the order and linearity of each equation. It also asks the reader to find the values of m that satisfy two ordinary differential equations, where y is defined as a function of m. Specifically, it asks the reader to find the values of m such that y=xm and y=emx cos(nx) are solutions to two given ODEs.
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The midterm exam document provides instructions and 6 problems to solve for partial differential equations (PDE). It instructs students to solve 4 of the 6 problems and clearly indicate which problems they wish marked. The problems cover determining if an ODE is exact and solving it, identifying linear and non-linear ODEs, performing a substitution to obtain a new linear ODE, modeling cake cooling with an exponential function, and solving an ODE using a substitution of the form u=Ax+By+C. Students have 1 hour and 15 minutes to complete 4 of the problems out of a possible 40 points.
Differential equations
The document discusses separable differential equations, which can be expressed as the product of a function of x and a function of y. An example separable differential equation is provided, along with steps to solve it by separating the variables. The solution yields an implicit relationship between y and x. A second example problem is presented to illustrate solving a separable differential equation with an initial condition.
Chapter 15
This document contains 20 multiple integral exercises with solutions. Some of the exercises involve calculating double integrals over specified regions, while others involve setting up approximations of double integrals using Riemann sums. Exercise 19 involves sketching solid regions in 3D space and Exercise 20 involves sketching surfaces defined by z=f(x,y).
difeq1beamer.pdf
The document defines differential equations and their order and degree. A differential equation contains a function and its derivatives. Ordinary differential equations have one independent variable, while partial differential equations have more. The order is the highest derivative, and the degree is the highest derivative treated as a variable. Solutions to differential equations are functions that satisfy the equation, with general solutions representing all possibilities and particular solutions being specific cases. Examples of differential equations and their solutions are provided.
Double integration
A double integral calculates the volume under a surface z=f(x,y) over a region R in the xy-plane, similar to how a single integral calculates area under a curve. To evaluate a double integral, integrate the inner integral first with respect to one variable (treating the other as a constant), then integrate the result with respect to the other variable. The order of integration can impact difficulty, but both must yield the same answer. Examples provided evaluate double integrals over rectangular and non-rectangular regions, demonstrating different integration order techniques.
11X1 T09 03 second derivative
The document discusses the second derivative and provides examples of taking the derivative of two functions. It shows that the second derivative of y=x^2 + 3 is 2x.
Tut 1
This document contains 11 questions regarding differential equations. It asks the student to classify equations as order, linear or non-linear; formulate differential equations by eliminating constants; solve specific differential equations; prove properties of curves; find general solutions by substitution; and determine conditions for exactness and solve exact differential equations.
Implicit differentiation
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
Sect1 4
1. The document provides examples of separable differential equations and their solutions. It emphasizes that looking for separable equations is the first step.
2. Twenty-seven examples of separable differential equations are given along with their solutions. Initial value problems are used to determine constants of integration in some cases.
3. Applications involving population growth, radioactive decay, bank accounts, and drug dosing are modeled with separable differential equations.
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