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
Since is constant with respect to , the derivative of with respect to is .
Step 1.4
Differentiate using the Power Rule which states that is where .
Step 1.5
Step 2
Find where .
Step 2.1
Differentiate with respect to .
Step 2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
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
Differentiate.
Step 8.2.1
By the Sum Rule, the derivative of with respect to is .
Step 8.2.2
Since is constant with respect to , 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
Differentiate using the function rule which states that the derivative of is .
Step 8.5
Simplify.
Step 8.5.1
Step 8.5.2
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
Combine the opposite terms in .
Step 9.1.2.1
Subtract from .
Step 9.1.2.2