# 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

Take the derivative of .

Rewrite the expression using the negative exponent rule .

Differentiate using the Quotient Rule which states that is where and .

Differentiate using the Constant Rule.

Multiply by .

Since is constant with respect to , the derivative of with respect to is .

Simplify the expression.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

Rewrite as .

Step 5

Substitute for and for in the original equation .

Step 6

Multiply each term in by to eliminate the fractions.

Multiply each term in by .

Simplify the left side.

Simplify each term.

Cancel the common factor of .

Move the leading negative in into the numerator.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Multiply by .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Subtract from .

Simplify .

Multiply by .

Multiply by .

Simplify the right side.

Rewrite using the commutative property of multiplication.

Multiply the exponents in .

Apply the power rule and multiply exponents, .

Multiply by .

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Subtract from .

Simplify .

The integrating factor is defined by the formula , where .

Set up the integration.

Apply the constant rule.

Remove the constant of integration.

Multiply each term by the integrating factor .

Multiply each term by .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

Reorder factors in .

Rewrite the left side as a result of differentiating a product.

Set up an integral on each side.

Integrate the left side.

Integrate the right side.

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

Let . Then , so . Rewrite using and .

Let . Find .

Differentiate .

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 .

Rewrite the problem using and .

Combine and .

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

The integral of with respect to is .

Simplify.

Replace all occurrences of with .

Divide each term in by and simplify.

Divide each term in by .

Simplify the left side.

Cancel the common factor of .

Cancel the common factor.

Divide by .

Simplify the right side.

Simplify each term.

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Multiply by .

Cancel the common factor.

Rewrite the expression.

Divide by .

Combine and .

Step 7

Substitute for .