Calculus Examples

Solve the Differential Equation (ds)/(dt)=16t(4t^2-1)^3 , s(1)=-5
,
Step 1
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
Apply the constant rule.
Step 2.3
Integrate the right side.
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Step 2.3.1
Since is constant with respect to , move out of the integral.
Step 2.3.2
Let . Then , so . Rewrite using and .
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Step 2.3.2.1
Let . Find .
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Step 2.3.2.1.1
Differentiate .
Step 2.3.2.1.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3.2.1.3
Evaluate .
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Step 2.3.2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.2.1.3.3
Multiply by .
Step 2.3.2.1.4
Differentiate using the Constant Rule.
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Step 2.3.2.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2.1.4.2
Add and .
Step 2.3.2.2
Rewrite the problem using and .
Step 2.3.3
Combine and .
Step 2.3.4
Since is constant with respect to , move out of the integral.
Step 2.3.5
Simplify.
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Step 2.3.5.1
Combine and .
Step 2.3.5.2
Cancel the common factor of and .
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Step 2.3.5.2.1
Factor out of .
Step 2.3.5.2.2
Cancel the common factors.
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Step 2.3.5.2.2.1
Factor out of .
Step 2.3.5.2.2.2
Cancel the common factor.
Step 2.3.5.2.2.3
Rewrite the expression.
Step 2.3.5.2.2.4
Divide by .
Step 2.3.6
By the Power Rule, the integral of with respect to is .
Step 2.3.7
Simplify.
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Step 2.3.7.1
Rewrite as .
Step 2.3.7.2
Simplify.
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Step 2.3.7.2.1
Combine and .
Step 2.3.7.2.2
Cancel the common factor of and .
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Step 2.3.7.2.2.1
Factor out of .
Step 2.3.7.2.2.2
Cancel the common factors.
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Step 2.3.7.2.2.2.1
Factor out of .
Step 2.3.7.2.2.2.2
Cancel the common factor.
Step 2.3.7.2.2.2.3
Rewrite the expression.
Step 2.3.8
Replace all occurrences of with .
Step 2.4
Group the constant of integration on the right side as .
Step 3
Use the initial condition to find the value of by substituting for and for in .
Step 4
Solve for .
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Step 4.1
Rewrite the equation as .
Step 4.2
Simplify each term.
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Step 4.2.1
Simplify each term.
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Step 4.2.1.1
One to any power is one.
Step 4.2.1.2
Multiply by .
Step 4.2.2
Subtract from .
Step 4.2.3
Raise to the power of .
Step 4.2.4
Combine and .
Step 4.3
Move all terms not containing to the right side of the equation.
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Step 4.3.1
Subtract from both sides of the equation.
Step 4.3.2
To write as a fraction with a common denominator, multiply by .
Step 4.3.3
Combine and .
Step 4.3.4
Combine the numerators over the common denominator.
Step 4.3.5
Simplify the numerator.
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Step 4.3.5.1
Multiply by .
Step 4.3.5.2
Subtract from .
Step 4.3.6
Move the negative in front of the fraction.
Step 5
Substitute for in and simplify.
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Step 5.1
Substitute for .
Step 5.2
Simplify each term.
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Step 5.2.1
Use the Binomial Theorem.
Step 5.2.2
Simplify each term.
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Step 5.2.2.1
Apply the product rule to .
Step 5.2.2.2
Raise to the power of .
Step 5.2.2.3
Multiply the exponents in .
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Step 5.2.2.3.1
Apply the power rule and multiply exponents, .
Step 5.2.2.3.2
Multiply by .
Step 5.2.2.4
Apply the product rule to .
Step 5.2.2.5
Multiply by by adding the exponents.
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Step 5.2.2.5.1
Move .
Step 5.2.2.5.2
Multiply by .
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Step 5.2.2.5.2.1
Raise to the power of .
Step 5.2.2.5.2.2
Use the power rule to combine exponents.
Step 5.2.2.5.3
Add and .
Step 5.2.2.6
Raise to the power of .
Step 5.2.2.7
Multiply the exponents in .
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Step 5.2.2.7.1
Apply the power rule and multiply exponents, .
Step 5.2.2.7.2
Multiply by .
Step 5.2.2.8
Multiply by .
Step 5.2.2.9
Apply the product rule to .
Step 5.2.2.10
Raise to the power of .
Step 5.2.2.11
Multiply the exponents in .
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Step 5.2.2.11.1
Apply the power rule and multiply exponents, .
Step 5.2.2.11.2
Multiply by .
Step 5.2.2.12
Multiply by .
Step 5.2.2.13
Raise to the power of .
Step 5.2.2.14
Multiply by .
Step 5.2.2.15
Multiply by .
Step 5.2.2.16
Raise to the power of .
Step 5.2.2.17
Multiply by .
Step 5.2.2.18
Raise to the power of .
Step 5.2.3
Apply the distributive property.
Step 5.2.4
Simplify.
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Step 5.2.4.1
Cancel the common factor of .
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Step 5.2.4.1.1
Factor out of .
Step 5.2.4.1.2
Cancel the common factor.
Step 5.2.4.1.3
Rewrite the expression.
Step 5.2.4.2
Cancel the common factor of .
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Step 5.2.4.2.1
Factor out of .
Step 5.2.4.2.2
Cancel the common factor.
Step 5.2.4.2.3
Rewrite the expression.
Step 5.2.4.3
Cancel the common factor of .
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Step 5.2.4.3.1
Factor out of .
Step 5.2.4.3.2
Cancel the common factor.
Step 5.2.4.3.3
Rewrite the expression.
Step 5.2.4.4
Cancel the common factor of .
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Step 5.2.4.4.1
Factor out of .
Step 5.2.4.4.2
Cancel the common factor.
Step 5.2.4.4.3
Rewrite the expression.
Step 5.2.4.5
Multiply by .
Step 5.3
Combine the numerators over the common denominator.
Step 5.4
Subtract from .
Step 5.5
Divide by .