# Calculus Examples

Solve the Differential Equation
Step 1
To solve the differential equation, let where is the exponent of .
Step 2
Solve the equation for .
Step 3
Take the derivative of with respect to .
Step 4
Take the derivative of with respect to .
Step 4.1
Take the derivative of .
Step 4.2
Rewrite the expression using the negative exponent rule .
Step 4.3
Differentiate using the Quotient Rule which states that is where and .
Step 4.4
Differentiate using the Constant Rule.
Step 4.4.1
Multiply by .
Step 4.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.4.3
Simplify the expression.
Step 4.4.3.1
Multiply by .
Step 4.4.3.2
Subtract from .
Step 4.4.3.3
Move the negative in front of the fraction.
Step 4.5
Rewrite as .
Step 5
Substitute for and for in the original equation .
Step 6
Solve the substituted differential equation.
Step 6.1
Multiply each term in by to eliminate the fractions.
Step 6.1.1
Multiply each term in by .
Step 6.1.2
Simplify the left side.
Step 6.1.2.1
Simplify each term.
Step 6.1.2.1.1
Cancel the common factor of .
Step 6.1.2.1.1.1
Move the leading negative in into the numerator.
Step 6.1.2.1.1.2
Factor out of .
Step 6.1.2.1.1.3
Cancel the common factor.
Step 6.1.2.1.1.4
Rewrite the expression.
Step 6.1.2.1.2
Multiply by .
Step 6.1.2.1.3
Multiply by .
Step 6.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 6.1.2.1.5
Multiply by by adding the exponents.
Step 6.1.2.1.5.1
Move .
Step 6.1.2.1.5.2
Use the power rule to combine exponents.
Step 6.1.2.1.5.3
Subtract from .
Step 6.1.2.1.6
Simplify .
Step 6.1.2.1.7
Multiply by .
Step 6.1.2.1.8
Multiply by .
Step 6.1.3
Simplify the right side.
Step 6.1.3.1
Rewrite using the commutative property of multiplication.
Step 6.1.3.2
Multiply the exponents in .
Step 6.1.3.2.1
Apply the power rule and multiply exponents, .
Step 6.1.3.2.2
Multiply by .
Step 6.1.3.3
Multiply by by adding the exponents.
Step 6.1.3.3.1
Move .
Step 6.1.3.3.2
Use the power rule to combine exponents.
Step 6.1.3.3.3
Subtract from .
Step 6.1.3.4
Simplify .
Step 6.2
The integrating factor is defined by the formula , where .
Step 6.2.1
Set up the integration.
Step 6.2.2
Apply the constant rule.
Step 6.2.3
Remove the constant of integration.
Step 6.3
Multiply each term by the integrating factor .
Step 6.3.1
Multiply each term by .
Step 6.3.2
Rewrite using the commutative property of multiplication.
Step 6.3.3
Multiply by by adding the exponents.
Step 6.3.3.1
Move .
Step 6.3.3.2
Use the power rule to combine exponents.
Step 6.3.3.3
Step 6.3.4
Reorder factors in .
Step 6.4
Rewrite the left side as a result of differentiating a product.
Step 6.5
Set up an integral on each side.
Step 6.6
Integrate the left side.
Step 6.7
Integrate the right side.
Step 6.7.1
Since is constant with respect to , move out of the integral.
Step 6.7.2
Let . Then , so . Rewrite using and .
Step 6.7.2.1
Let . Find .
Step 6.7.2.1.1
Differentiate .
Step 6.7.2.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.7.2.1.3
Differentiate using the Power Rule which states that is where .
Step 6.7.2.1.4
Multiply by .
Step 6.7.2.2
Rewrite the problem using and .
Step 6.7.3
Combine and .
Step 6.7.4
Since is constant with respect to , move out of the integral.
Step 6.7.5
The integral of with respect to is .
Step 6.7.6
Simplify.
Step 6.7.7
Replace all occurrences of with .
Step 6.8
Divide each term in by and simplify.
Step 6.8.1
Divide each term in by .
Step 6.8.2
Simplify the left side.
Step 6.8.2.1
Cancel the common factor of .
Step 6.8.2.1.1
Cancel the common factor.
Step 6.8.2.1.2
Divide by .
Step 6.8.3
Simplify the right side.
Step 6.8.3.1
Simplify each term.
Step 6.8.3.1.1
Cancel the common factor of and .
Step 6.8.3.1.1.1
Factor out of .
Step 6.8.3.1.1.2
Cancel the common factors.
Step 6.8.3.1.1.2.1
Multiply by .
Step 6.8.3.1.1.2.2
Cancel the common factor.
Step 6.8.3.1.1.2.3
Rewrite the expression.
Step 6.8.3.1.1.2.4
Divide by .
Step 6.8.3.1.2
Combine and .
Step 7
Substitute for .