# Calculus Examples

Verify the Existence and Uniqueness of Solutions for the Differential Equation
,
Step 1
Assume .
Step 2
Check if the function is continuous in the neighborhood of .
Step 2.1
Substitute values into .
Step 2.1.1
Substitute for .
Step 2.1.2
Substitute for .
Step 2.2
Since there is no log with negative or zero argument, no even radical with zero or negative radicand, and no fraction with zero in the denominator, the function is continuous on an open interval around the value of .
Continuous
Continuous
Step 3
Find the partial derivative with respect to .
Step 3.1
Set up the partial derivative.
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the Power Rule which states that is where .
Step 3.4
Multiply by .
Step 4
Check if the partial derivative with respect to is continuous in the neighborhood of .
Step 4.1
Since there is no log with negative or zero argument, no even radical with zero or negative radicand, and no fraction with zero in the denominator, the function is continuous on an open interval around the value of .
Continuous
Continuous
Step 5
Both the function and its partial derivative with respect to are continuous on an open interval around the value of .
One unique solution