# Calculus Examples

Find the Values of r that Satisfy the Differential Equation
,
Step 1
Find .
Step 1.1
Differentiate both sides of the equation.
Step 1.2
The derivative of with respect to is .
Step 1.3
Differentiate the right side of the equation.
Step 1.3.1
Differentiate using the chain rule, which states that is where and .
Step 1.3.1.1
To apply the Chain Rule, set as .
Step 1.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.3.1.3
Replace all occurrences of with .
Step 1.3.2
Differentiate.
Step 1.3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2.2
Differentiate using the Power Rule which states that is where .
Step 1.3.2.3
Simplify the expression.
Step 1.3.2.3.1
Multiply by .
Step 1.3.2.3.2
Reorder factors in .
Step 1.4
Reform the equation by setting the left side equal to the right side.
Step 2
Find .
Step 2.1
Set up the derivative.
Step 2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Differentiate using the chain rule, which states that is where and .
Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
Raise to the power of .
Step 2.6
Raise to the power of .
Step 2.7
Use the power rule to combine exponents.
Step 2.8
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Multiply by .
Step 3
Substitute into the given differential equation.
Step 4
Substitute for .
Step 5
Solve for .
Step 5.1
Divide each term in by and simplify.
Step 5.1.1
Divide each term in by .
Step 5.1.2
Simplify the left side.
Step 5.1.2.1
Cancel the common factor of .
Step 5.1.2.1.1
Cancel the common factor.
Step 5.1.2.1.2
Rewrite the expression.
Step 5.1.2.2
Cancel the common factor of .
Step 5.1.2.2.1
Cancel the common factor.
Step 5.1.2.2.2
Divide by .
Step 5.1.3
Simplify the right side.
Step 5.1.3.1
Cancel the common factor of .
Step 5.1.3.1.1
Cancel the common factor.
Step 5.1.3.1.2
Rewrite the expression.
Step 5.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.3
Simplify .
Step 5.3.1
Rewrite as .
Step 5.3.2
Any root of is .
Step 5.3.3
Simplify the denominator.
Step 5.3.3.1
Rewrite as .
Step 5.3.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 5.4
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.4.1
First, use the positive value of the to find the first solution.
Step 5.4.2
Next, use the negative value of the to find the second solution.
Step 5.4.3
The complete solution is the result of both the positive and negative portions of the solution.