Calculus Examples

Solve the Differential Equation
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 with respect to .
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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.
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Multiply by .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Rewrite as .
Step 5
Substitute for in the original equation .
Step 6
Substitute for in the original equation .
Step 7
Solve the substituted differential equation.
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Substitute for in the original equation .
Substitute for in the original equation .
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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Simplify each term.
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Cancel the common factor of .
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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.
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Move .
Use the power rule to combine exponents.
Subtract from .
Simplify .
Multiply by .
Multiply by .
Simplify the right side.
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Rewrite using the commutative property of multiplication.
Multiply the exponents in .
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Apply the power rule and multiply exponents, .
Multiply by .
Multiply by by adding the exponents.
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Move .
Use the power rule to combine exponents.
Subtract from .
Simplify .
The integrating factor is defined by the formula , where .
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Set up the integration.
Apply the constant rule.
Remove the constant of integration.
Multiply each term by the integrating factor .
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Multiply each term by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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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.
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Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
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Let . Find .
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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.
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Divide 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.
Divide by .
Simplify the right side.
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Simplify each term.
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Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Multiply by .
Cancel the common factor.
Rewrite the expression.
Divide by .
Combine and .
Step 8
Substitute for .
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