# Calculus Examples

Solve the Differential Equation
Step 1
Find where .
Step 1.1
Differentiate with respect to .
Step 1.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3
Evaluate .
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Differentiate using the Power Rule which states that is where .
Step 2
Find where .
Step 2.1
Differentiate with respect to .
Step 2.2
Differentiate.
Step 2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Evaluate .
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Differentiate using the Constant Rule.
Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Step 3
Check that .
Step 3.1
Substitute for and for .
Step 3.2
Since the two sides have been shown to be equivalent, the equation is an identity.
is an identity.
is an identity.
Step 4
Set equal to the integral of .
Step 5
Integrate to find .
Step 5.1
Split the single integral into multiple integrals.
Step 5.2
Since is constant with respect to , move out of the integral.
Step 5.3
By the Power Rule, the integral of with respect to is .
Step 5.4
Apply the constant rule.
Step 5.5
Combine and .
Step 5.6
Simplify.
Step 6
Since the integral of will contain an integration constant, we can replace with .
Step 7
Set .
Step 8
Find .
Step 8.1
Differentiate with respect to .
Step 8.2
By the Sum Rule, the derivative of with respect to is .
Step 8.3
Evaluate .
Step 8.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 8.3.2
Differentiate using the Power Rule which states that is where .
Step 8.3.3
Multiply by .
Step 8.4
Evaluate .
Step 8.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 8.4.2
Differentiate using the Power Rule which states that is where .
Step 8.4.3
Move to the left of .
Step 8.5
Differentiate using the function rule which states that the derivative of is .
Step 8.6
Reorder terms.
Step 9
Solve for .
Step 9.1
Move all terms not containing to the right side of the equation.
Step 9.1.1
Subtract from both sides of the equation.
Step 9.1.2
Subtract from both sides of the equation.
Step 9.1.3
Combine the opposite terms in .
Step 9.1.3.1
Subtract from .
Step 9.1.3.2
Step 9.1.3.3
Subtract from .
Step 9.1.3.4
Step 10
Find the antiderivative of to find .
Step 10.1
Integrate both sides of .
Step 10.2
Evaluate .
Step 10.3
Apply the constant rule.
Step 11
Substitute for in .