Calculus Examples

Solve the Differential Equation
Step 1
The integrating factor is defined by the formula , where .
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Step 1.1
Set up the integration.
Step 1.2
Apply the constant rule.
Step 1.3
Remove the constant of integration.
Step 2
Multiply each term by the integrating factor .
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Step 2.1
Multiply each term by .
Step 2.2
Multiply by by adding the exponents.
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Step 2.2.1
Use the power rule to combine exponents.
Step 2.2.2
Add and .
Step 2.3
Reorder factors in .
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
Integrate the right side.
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Step 6.1
Let . Then , so . Rewrite using and .
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Step 6.1.1
Let . Find .
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Step 6.1.1.1
Differentiate .
Step 6.1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.1.3
Differentiate using the Power Rule which states that is where .
Step 6.1.1.4
Multiply by .
Step 6.1.2
Rewrite the problem using and .
Step 6.2
Combine and .
Step 6.3
Since is constant with respect to , move out of the integral.
Step 6.4
The integral of with respect to is .
Step 6.5
Simplify.
Step 6.6
Replace all occurrences of with .
Step 7
Divide each term in by and simplify.
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Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
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Step 7.2.1
Cancel the common factor of .
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Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.3
Simplify the right side.
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Step 7.3.1
Simplify each term.
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Step 7.3.1.1
Cancel the common factor of and .
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Step 7.3.1.1.1
Factor out of .
Step 7.3.1.1.2
Cancel the common factors.
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Step 7.3.1.1.2.1
Multiply by .
Step 7.3.1.1.2.2
Cancel the common factor.
Step 7.3.1.1.2.3
Rewrite the expression.
Step 7.3.1.1.2.4
Divide by .
Step 7.3.1.2
Combine and .
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