Calculus Examples

Solve the Differential Equation 3/2(dv)/(dt)=13.6-1/2v
Step 1
Separate the variables.
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Step 1.1
Multiply both sides by .
Step 1.2
Cancel the common factor of .
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Step 1.2.1
Cancel the common factor.
Step 1.2.2
Rewrite the expression.
Step 1.3
Remove unnecessary parentheses.
Step 1.4
Rewrite the equation.
Step 2
Integrate both sides.
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Step 2.1
Set up an integral on each side.
Step 2.2
Integrate the left side.
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Step 2.2.1
Simplify.
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Step 2.2.1.1
Combine and .
Step 2.2.1.2
Multiply by .
Step 2.2.1.3
Move to the left of .
Step 2.2.2
Since is constant with respect to , move out of the integral.
Step 2.2.3
Let . Then , so . Rewrite using and .
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Step 2.2.3.1
Let . Find .
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Step 2.2.3.1.1
Differentiate .
Step 2.2.3.1.2
Differentiate.
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Step 2.2.3.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.3.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.1.3
Evaluate .
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Step 2.2.3.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3.1.3.3
Multiply by .
Step 2.2.3.1.4
Subtract from .
Step 2.2.3.2
Rewrite the problem using and .
Step 2.2.4
Simplify.
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Step 2.2.4.1
Move the negative in front of the fraction.
Step 2.2.4.2
Dividing two negative values results in a positive value.
Step 2.2.4.3
Multiply by the reciprocal of the fraction to divide by .
Step 2.2.4.4
Multiply by .
Step 2.2.4.5
Multiply by .
Step 2.2.4.6
Combine and .
Step 2.2.4.7
Move the negative in front of the fraction.
Step 2.2.5
Since is constant with respect to , move out of the integral.
Step 2.2.6
Since is constant with respect to , move out of the integral.
Step 2.2.7
Simplify.
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Step 2.2.7.1
Multiply by .
Step 2.2.7.2
Combine and .
Step 2.2.7.3
Multiply by .
Step 2.2.7.4
Cancel the common factor of and .
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Step 2.2.7.4.1
Factor out of .
Step 2.2.7.4.2
Cancel the common factors.
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Step 2.2.7.4.2.1
Factor out of .
Step 2.2.7.4.2.2
Cancel the common factor.
Step 2.2.7.4.2.3
Rewrite the expression.
Step 2.2.7.4.2.4
Divide by .
Step 2.2.8
The integral of with respect to is .
Step 2.2.9
Simplify.
Step 2.2.10
Replace all occurrences of with .
Step 2.2.11
Reorder terms.
Step 2.3
Apply the constant rule.
Step 2.4
Group the constant of integration on the right side as .
Step 3
Solve for .
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Step 3.1
Divide each term in by and simplify.
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Step 3.1.1
Divide each term in by .
Step 3.1.2
Simplify the left side.
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Step 3.1.2.1
Cancel the common factor of .
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Step 3.1.2.1.1
Cancel the common factor.
Step 3.1.2.1.2
Divide by .
Step 3.1.2.2
Combine and .
Step 3.1.3
Simplify the right side.
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Step 3.1.3.1
Simplify each term.
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Step 3.1.3.1.1
Move the negative in front of the fraction.
Step 3.1.3.1.2
Move the negative in front of the fraction.
Step 3.2
To solve for , rewrite the equation using properties of logarithms.
Step 3.3
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 3.4
Solve for .
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Step 3.4.1
Rewrite the equation as .
Step 3.4.2
Remove the absolute value term. This creates a on the right side of the equation because .
Step 3.4.3
Subtract from both sides of the equation.
Step 3.4.4
Multiply both sides of the equation by .
Step 3.4.5
Simplify both sides of the equation.
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Step 3.4.5.1
Simplify the left side.
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Step 3.4.5.1.1
Simplify .
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Step 3.4.5.1.1.1
Cancel the common factor of .
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Step 3.4.5.1.1.1.1
Move the leading negative in into the numerator.
Step 3.4.5.1.1.1.2
Factor out of .
Step 3.4.5.1.1.1.3
Cancel the common factor.
Step 3.4.5.1.1.1.4
Rewrite the expression.
Step 3.4.5.1.1.2
Multiply.
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Step 3.4.5.1.1.2.1
Multiply by .
Step 3.4.5.1.1.2.2
Multiply by .
Step 3.4.5.2
Simplify the right side.
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Step 3.4.5.2.1
Simplify .
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Step 3.4.5.2.1.1
Apply the distributive property.
Step 3.4.5.2.1.2
Multiply by .
Step 3.4.6
Reorder and .
Step 4
Group the constant terms together.
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Step 4.1
Simplify the constant of integration.
Step 4.2
Reorder terms.
Step 4.3
Rewrite as .
Step 4.4
Reorder and .
Step 4.5
Combine constants with the plus or minus.