Calculus Examples

Verify the Differential Equation Solution
,
Step 1
Find .
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Differentiate both sides of the equation.
The derivative of with respect to is .
Differentiate the right side of the equation.
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Differentiate.
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Since is constant with respect to , the derivative of with respect to is .
Multiply by .
Differentiate using the Power Rule which states that is where .
Multiply by .
Reform the equation by setting the left side equal to the right side.
Step 2
Substitute into the given differential equation.
Step 3
Simplify.
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Multiply by .
Add and .
Step 4
The given solution satisfies the given differential equation.
is a solution to
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