Question

In Problems, $\mathbf{17}\mathbf{-}\mathbf{24}$ determine a region of the xy-plane for which

the given differential equation would have a unique solution whose

graph passes through a point $\mathbf{(}{\mathit{x}}_{\mathbf{0}}\mathbf{,}{\mathit{y}}_{\mathbf{0}}\mathbf{)}$ in the region.

$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{-}\mathit{y}\mathbf{=}\mathit{x}$

Short answer

The required region where the differential equation have unique solution is everywhere in xy-plane.

Step-by-step solution

Note the given data

Given differential equation, $\frac{dy}{dx}-y=x$

Add $y$both the sides and we get the function $f(x,y)=x+y$

It contains an interior point $(x_0,y_0)$in the graph

Finding the continuity of the function

The given function $f(x,y)=x+y$

The function $f(x,y)=x+y$is continuous everywhere in xy-plane.

Finding the partial derivative

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

$\frac{\delta f}{\delta y}=\frac{\delta }{\delta y}\left(x+y\right)$

$=1$

This partial derivative is continuous everywhere in xy-plane.

Finding the required region

From given data $(x_0,y_0)$is an interior point.

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.