Calculus Examples

Solve the Differential Equation
Step 1
Let . Substitute for .
Step 2
Solve for .
Step 3
Use the product rule to find the derivative of with respect to .
Step 4
Substitute for .
Step 5
Solve the substituted differential equation.
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Separate the variables.
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Solve for .
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Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Combine the opposite terms in .
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Subtract from .
Add and .
Divide each term in by and simplify.
Multiply both sides by .
Simplify.
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Rewrite using the commutative property of multiplication.
Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Rewrite the equation.
Integrate both sides.
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Set up an integral on each side.
Integrate the left side.
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Apply basic rules of exponents.
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Move out of the denominator by raising it to the power.
Multiply the exponents in .
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Apply the power rule and multiply exponents, .
Multiply by .
By the Power Rule, the integral of with respect to is .
Rewrite as .
Integrate the right side.
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Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Simplify.
Group the constant of integration on the right side as .
Solve for .
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Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
The LCM of one and any expression is the expression.
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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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Simplify the right side.
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Reorder factors in .
Solve the equation.
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Rewrite the equation as .
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Rewrite as .
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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Move the negative in front of the fraction.
Factor out of .
Factor out of .
Factor out of .
Simplify the expression.
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Rewrite as .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
Simplify the constant of integration.
Step 6
Substitute for .
Step 7
Solve for .
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Multiply both sides by .
Simplify.
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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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Combine and .
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