Chapter 1: Q4RP (page 1)
In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.
⦁ The Independent variable is t.
⦁ The dependent variable is x.
⦁ The equation is Non-Linear.
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In problems Use Euler’s method to approximate the solution to the given initial value problem at the points x = 0.1, 0.2, 0.3, 0.4, and 0.5, using steps of size 0.1 (h = 0.1).
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Implicit Function Theorem. Let have continuous first partial derivatives in the rectanglecontaining the pointlocalid="1664009358887" . If and the partial derivative, then there exists a differentiable function , defined in some interval,that satisfies G for allforall .
The implicit function theorem gives conditions under which the relationship implicitly defines yas a function of x. Use the implicit function theorem to show that the relationship given in Example 4, defines y implicitly as a function of x near the point.
Consider the differential equation for the population p (in thousands) of a certain species at time t.
⦁ Sketch the direction field by using either a computer software package or the method of isoclines.
⦁ If the initial population is 4000 [that is, ], what can you say about the limiting population
⦁ If , what is
⦁ If , what is
⦁ Can a population of 900 ever increase to 1100?
In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.
,
In Problems 3–8, determine whether the given function is a solution to the given differential equation.
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