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Chapter 8: Q14P (page 398)

In problems 13 to 15, find a solution(or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.

14. Problem 8.

Short Answer

Expert verified

Answer

The solution is y=0.

Step by step solution

01

Given information

The given differential equation is y'+2xy2=0

02

Definition of differential equation

A differential equation is an equation that contains at least onederivative of an unknown function, either an ordinary derivative or a partial derivative.

03

Solve the differential equation

Separate the variables in problem 8 to get

dyy2=-2xdx.

The general solution of this differential equation is

y=1x2+C.

Now, dyy2is not valid for y=0.

y=0is a solution of this differential equation that cannot be obtained by any choice of

C.

Therefore, the solution is y=0.

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Most popular questions from this chapter

In Problem 33 to 38, solve the given differential equations by using the principle of superposition [see the solution of equation (6.29)]. For example, in Problem 33, solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus, a polynomial of any degree is kept together in one bracket.

(D-1)2y=4ex+(1-x)(e2x-1)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

y"-4y=4e2t,鈭�y0=0,y'0=1

Heat is escaping at a constant rate [dQdtin (1.1)is constant] through the walls of a long cylindrical pipe. Find the temperature T at a distance r from the axis of the cylinder if the inside wall has radius r=1and temperature T=100and the outside wall has r=2and T=0

Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after (3.9), and Example 1.

y'cosx+y=cos2x

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from y'for the original curves; this constant takes different values for different curves of the original family, and you want an expression for y'which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations 2.10to 2.12.

(y-1)2=x2+k
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