# Calculus Examples

Solve the Differential Equation
Step 1
The integrating factor is defined by the formula , where .
Step 1.1
Set up the integration.
Step 1.2
Integrate .
Step 1.2.1
Since is constant with respect to , move out of the integral.
Step 1.2.2
The integral of with respect to is .
Step 1.2.3
Simplify.
Step 1.3
Remove the constant of integration.
Step 1.4
Use the logarithmic power rule.
Step 1.5
Exponentiation and log are inverse functions.
Step 1.6
Rewrite the expression using the negative exponent rule .
Step 2
Multiply each term by the integrating factor .
Step 2.1
Multiply each term by .
Step 2.2
Simplify each term.
Step 2.2.1
Combine and .
Step 2.2.2
Rewrite using the commutative property of multiplication.
Step 2.2.3
Combine and .
Step 2.2.4
Multiply .
Step 2.2.4.1
Multiply by .
Step 2.2.4.2
Raise to the power of .
Step 2.2.4.3
Raise to the power of .
Step 2.2.4.4
Use the power rule to combine exponents.
Step 2.2.4.5
Step 2.3
Rewrite using the commutative property of multiplication.
Step 2.4
Combine and .
Step 2.5
Cancel the common factor of .
Step 2.5.1
Cancel the common factor.
Step 2.5.2
Rewrite the expression.
Step 3
Rewrite the left side as a result of differentiating a product.
Step 4
Set up an integral on each side.
Step 5
Integrate the left side.
Step 6
Apply the constant rule.
Step 7
Solve for .
Step 7.1
Combine and .
Step 7.2
Multiply both sides by .
Step 7.3
Simplify.
Step 7.3.1
Simplify the left side.
Step 7.3.1.1
Cancel the common factor of .
Step 7.3.1.1.1
Cancel the common factor.
Step 7.3.1.1.2
Rewrite the expression.
Step 7.3.2
Simplify the right side.
Step 7.3.2.1
Simplify .
Step 7.3.2.1.1
Apply the distributive property.
Step 7.3.2.1.2
Multiply by by adding the exponents.
Step 7.3.2.1.2.1
Move .
Step 7.3.2.1.2.2
Multiply by .
Step 7.3.2.1.3
Reorder and .