# Calculus Examples

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

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

Step 6.1

Rewrite the differential equation as .

Step 6.1.1

Rewrite the equation as .

Step 6.1.1.1

Multiply each term in by to eliminate the fractions.

Step 6.1.1.1.1

Multiply each term in by .

Step 6.1.1.1.2

Simplify the left side.

Step 6.1.1.1.2.1

Simplify each term.

Step 6.1.1.1.2.1.1

Cancel the common factor of .

Step 6.1.1.1.2.1.1.1

Move the leading negative in into the numerator.

Step 6.1.1.1.2.1.1.2

Factor out of .

Step 6.1.1.1.2.1.1.3

Cancel the common factor.

Step 6.1.1.1.2.1.1.4

Rewrite the expression.

Step 6.1.1.1.2.1.2

Multiply by .

Step 6.1.1.1.2.1.3

Multiply by .

Step 6.1.1.1.2.1.4

Multiply by by adding the exponents.

Step 6.1.1.1.2.1.4.1

Move .

Step 6.1.1.1.2.1.4.2

Use the power rule to combine exponents.

Step 6.1.1.1.2.1.4.3

Subtract from .

Step 6.1.1.1.2.1.5

Simplify .

Step 6.1.1.1.2.1.6

Multiply by .

Step 6.1.1.1.3

Simplify the right side.

Step 6.1.1.1.3.1

Rewrite using the commutative property of multiplication.

Step 6.1.1.1.3.2

Multiply the exponents in .

Step 6.1.1.1.3.2.1

Apply the power rule and multiply exponents, .

Step 6.1.1.1.3.2.2

Multiply by .

Step 6.1.1.1.3.3

Multiply by by adding the exponents.

Step 6.1.1.1.3.3.1

Move .

Step 6.1.1.1.3.3.2

Use the power rule to combine exponents.

Step 6.1.1.1.3.3.3

Subtract from .

Step 6.1.1.1.3.4

Simplify .

Step 6.1.1.2

Reorder terms.

Step 6.1.2

Factor out of .

Step 6.1.3

Reorder and .

Step 6.2

The integrating factor is defined by the formula , where .

Step 6.2.1

Set up the integration.

Step 6.2.2

Integrate .

Step 6.2.2.1

Since is constant with respect to , move out of the integral.

Step 6.2.2.2

By the Power Rule, the integral of with respect to is .

Step 6.2.2.3

Simplify the answer.

Step 6.2.2.3.1

Rewrite as .

Step 6.2.2.3.2

Simplify.

Step 6.2.2.3.2.1

Combine and .

Step 6.2.2.3.2.2

Cancel the common factor of and .

Step 6.2.2.3.2.2.1

Factor out of .

Step 6.2.2.3.2.2.2

Cancel the common factors.

Step 6.2.2.3.2.2.2.1

Factor out of .

Step 6.2.2.3.2.2.2.2

Cancel the common factor.

Step 6.2.2.3.2.2.2.3

Rewrite the expression.

Step 6.2.2.3.2.2.2.4

Divide by .

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

Rewrite using the commutative property of multiplication.

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

Simplify.

Step 6.7.3.1

Move the negative in front of the fraction.

Step 6.7.3.2

Combine and .

Step 6.7.3.3

Combine and .

Step 6.7.4

Since is constant with respect to , move out of the integral.

Step 6.7.5

Simplify.

Step 6.7.5.1

Multiply by .

Step 6.7.5.2

Multiply by .

Step 6.7.6

Since is constant with respect to , move out of the integral.

Step 6.7.7

Integrate by parts using the formula , where and .

Step 6.7.8

Simplify.

Step 6.7.8.1

Combine and .

Step 6.7.8.2

Combine and .

Step 6.7.8.3

Move to the left of .

Step 6.7.8.4

Move to the left of .

Step 6.7.8.5

Combine and .

Step 6.7.8.6

Combine and .

Step 6.7.8.7

Move to the left of .

Step 6.7.8.8

Move to the left of .

Step 6.7.9

Since is constant with respect to , move out of the integral.

Step 6.7.10

Simplify.

Step 6.7.10.1

Multiply by .

Step 6.7.10.2

Multiply by .

Step 6.7.11

Since is constant with respect to , move out of the integral.

Step 6.7.12

The integral of with respect to is .

Step 6.7.13

Simplify.

Step 6.7.13.1

Rewrite as .

Step 6.7.13.2

Simplify.

Step 6.7.13.2.1

Combine and .

Step 6.7.13.2.2

Combine and .

Step 6.7.13.2.3

Move to the left of .

Step 6.7.13.2.4

Move to the left of .

Step 6.7.13.2.5

Combine and .

Step 6.7.13.2.6

Combine and .

Step 6.7.13.2.7

Add and .

Step 6.7.13.2.8

Multiply by .

Step 6.7.13.2.9

Add and .

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 7

Substitute for .