Calculus Examples

Solve the Differential Equation
Step 1
The integrating factor is defined by the formula , where .
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Set up the integration.
Integrate .
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Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Simplify.
Remove the constant of integration.
Use the logarithmic power rule.
Exponentiation and log are inverse functions.
Rewrite the expression using the negative exponent rule .
Step 2
Multiply each term by the integrating factor .
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Multiply each term by .
Simplify each term.
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Combine and .
Rewrite using the commutative property of multiplication.
Combine and .
Multiply .
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Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite using the commutative property of multiplication.
Combine and .
Cancel the common factor of .
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Cancel the common factor.
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 .
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Combine and .
Multiply each term in by to eliminate the fractions.
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Multiply each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Simplify the right side.
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Multiply by by adding the exponents.
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Move .
Multiply by .
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