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In Problems, 17-24 determine a region of the xy-plane for which

the given differential equation would have a unique solution whose

graph passes through a point (x0,y0)in the region.

xdydx=y

Short Answer

Expert verified

The required region where the differential equation have unique solution is x<0 or x>0.

Step by step solution

01

Note the given data

Given differential equation, xdydx=y.

Dividing x from both the sides and we get the function f(x,y)=yx.

It contains an interior point(x0,y0) in the graph

02

Finding the continuity of the function

The given function f(x,y)=yx

The function f(x,y)=yxis continuous everywhere in xy-plane except x0

i.e., {(x,y):x0}.

03

Finding the partial derivative

Finding the partial derivative of the given function with respect to y

δfδy=δδyyx

=1x

This partial derivative is continuous when xnot equal zero. i.e., x<0or x>0.

04

Finding the required region

From given data (x0,y0)is an interior point.

So, we get x00

From the theorem of Existence of a unique solution, we get

The region of the unique solution exists of the given differential equation is everywhere in xy-plane except x equal to zero.

Hence, the region is x<0or x>0.

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