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Chapter 1: Q12RP (page 30)

The initial value problem \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{y}}^{\frac{{\bf{2}}}{{\bf{3}}}}}{\bf{,}}\;{\bf{y(2) = 1}}{{\bf{0}}^{{\bf{ - 100}}}}\]has a unique solution in some open interval around x = 2.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Finding partial derivatives

Since\[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{y}}^{\frac{{\bf{2}}}{{\bf{3}}}}}\]

Then\[\frac{{\partial {\bf{f}}}}{{\partial {\bf{y}}}}{\bf{ = }}\frac{{\bf{2}}}{{\sqrt[{\bf{3}}]{{\bf{y}}}}}\]

02

Checking the final result

The given function is continuous at x = 2.So the function is continuous at some open interval x = 2.

Therefore, the statement is true.

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Most popular questions from this chapter

Use Euler鈥檚 method with step size h = 0.1 to approximate the solution to the initial value problem

y'=x-y2, y (1) = 0 at the points x=1.1,1.2,1.3,1.4and1.5.

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 13-19,find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

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

y3d2xdy2+3x-8y-1=0

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