Calculus Examples Differential Equations

Verify the Existence and Uniqueness of Solutions for the Differential Equation

$\frac{d y}{d x} = \sqrt{x - y}$ , $(1, 1)$

Step 1

Assume $\frac{d y}{d x} = f (x, y)$ .

Step 2

Check if the function is continuous in the neighborhood of $(1, 1)$ .

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Step 2.1

Substitute $(1, 1)$ values into $\frac{d y}{d x} = \sqrt{x - y}$ .

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Step 2.1.1

Substitute $1$ for $x$ .

$\sqrt{1 - y}$

Step 2.1.2

Substitute $1$ for $y$ .

$\sqrt{1 - 1}$

Step 2.1.3

Subtract $1$ from $1$ .

$\sqrt{0}$

Step 2.2

There is an even radical with a zero as the radicand, which means that the function is not continuous on an open interval around the $x$ value of $(1, 1)$ .

Not continuous

Step 3

The function is not continuous on an open interval around the $x$ value of $(1, 1)$ .

A solution is not guaranteed

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Verify the Existence and Uniqueness of Solutions for the Differential Equation

Differential Equations

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