# Calculus Examples

Use the Initial Value to Solve for c
, ,
Step 1
Verify that the given solution satisfies the differential equation.
Step 1.1
Find .
Step 1.1.1
Differentiate both sides of the equation.
Step 1.1.2
The derivative of with respect to is .
Step 1.1.3
Differentiate the right side of the equation.
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the chain rule, which states that is where and .
Step 1.1.3.2.1
To apply the Chain Rule, set as .
Step 1.1.3.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.3.2.3
Replace all occurrences of with .
Step 1.1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.1.3.4
Simplify.
Step 1.1.3.4.1
Reorder the factors of .
Step 1.1.3.4.2
Reorder factors in .
Step 1.1.4
Reform the equation by setting the left side equal to the right side.
Step 1.2
Substitute into the given differential equation.
Step 1.3
Reorder factors in .
Step 1.4
The given solution satisfies the given differential equation.
is a solution to
is a solution to
Step 2
Substitute in the initial condition.
Step 3
Solve for .
Step 3.1
Rewrite the equation as .
Step 3.2
Divide each term in by and simplify.
Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
Step 3.2.2.1
Cancel the common factor of .
Step 3.2.2.1.1
Cancel the common factor.
Step 3.2.2.1.2
Divide by .
Step 3.2.3
Simplify the right side.
Step 3.2.3.1
Simplify the denominator.
Step 3.2.3.1.1
Raising to any positive power yields .
Step 3.2.3.1.2
Anything raised to is .
Step 3.2.3.2
Divide by .