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Chapter 1: Q3RP (page 1)

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

y3d2xdy2+3x-8y-1=0

Short Answer

Expert verified

猞� The Independent variable is y.

猞� The dependent variable is x.

猞� The equation is Linear.

Step by step solution

01

Identifying the dependent and independent variables

Clearly, the independent variable is y while the dependent variable is x.

02

Determining whether the equation is linear or nonlinear

Since the dependent variable x and its derivatives appear in additive combinations of their first powers, the equation is linear.

Hence, the Independent variable is y. The dependent variable is x. The equation is Linear.

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Most popular questions from this chapter

(a) For the initial value problem (12) of Example 9. Show that 蠒1(x)=0 and蠒2(x)=(x-2)3are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problemy'=3y23,y(0)=10-7have a unique solution in a neighbourhood ofx=0?

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.

x2-sin(x+y)=1,dydx=2xsec(x+y)-1

Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length l(t)of the wire varies with time in some predetermined fashion. If

U(t) is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem l2(t)胃''(t)+2l(t)l'(t)胃'(t)+gl(t)sin(胃(t))=0;胃(0)=胃o,胃'(0)=胃1where g is the acceleration due to gravity. Assume that l(t)=lo+l1cos(蝇迟-蠒)where l1is much smaller than lo. (This might be a model for a person on a swing, where the pumping action changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take g = 1. Using the Runge鈥� Kutta algorithm with h = 0.1, study the motion of the pendulum when 胃o=0.05,胃1=0,lo=1,l1=0.1,蝇=1,蠒=0.02. In particular, does the pendulum ever attain an angle greater in absolute value than the initial angle 胃o?

In Problems 14鈥�24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge鈥揔utta algorithm. (At the instructor鈥檚 discretion, other algorithms may be used.)鈥�

Using the vectorized Runge鈥揔utta algorithm, approximate the solution to the initial value problem

dudx=3u-4v;u(0)=1'dvdx=2u-3v;v(0)=1

at x = 1. Starting with h=1, continue halving the step size until two successive approximations of u(1)and v(1) differ by at most 0.001.

The logistic equation for the population (in thousands) of a certain species is given by dpdt=3p-2p2.

猞� Sketch the direction field by using either a computer software package or the method of isoclines.

猞� If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?

猞� If p(0)=0.8, what is limt鈫�+鈭�p(t)?

猞� Can a population of 2000 ever decline to 800?
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