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Chapter 1: Q27 E (page 14)

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

ydydx=x,y(1)=0

Short Answer

Expert verified

The hypotheses of Theorem 1 are not satisfied.

The initial value problem does not have a unique solution.

Step by step solution

01

Finding the partial derivative of the given relation concerning y

Here,fx,y=xyand鈭�f鈭�y=-xy2

02

Determining whether Theorem 1 implies the existence of a unique solution or not

From Step 1, we find that鈭�f鈭�yis not continuous or even defined when y=0. Consequently, there is no rectangle containing the point 1,0, in which bothfx,y and鈭�f鈭�y are continuous. Because the hypotheses of Theorem 1 do not hold, we cannot use Theorem 1 to determine whether the given initial value problem does or does not have a unique solution. It turns out that this initial value problem has more than one solution.

Hence, Theorem 1 implies that the given initial value problem does not have a unique solution.

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