Calculus Examples

Solve the Differential Equation
Step 1
Find where .
Differentiate with respect to .
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Power Rule which states that is where .
Step 2
Find where .
Differentiate with respect to .
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
Step 3
Check that .
Substitute for and for .
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 .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Apply the constant rule.
Combine and .
Simplify.
Step 6
Since the integral of will contain an integration constant, we can replace with .
Step 7
Set .
Step 8
Find .
Differentiate with respect to .
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Move to the left of .
Differentiate using the function rule which states that the derivative of is .
Reorder terms.
Step 9
Solve for .
Move all terms not containing to the right side of the equation.
Step 10
Find the antiderivative of to find .
Integrate both sides of .
Evaluate .
Apply the constant rule.
Step 11
Substitute for in .