Calculus Examples

Solve the Differential Equation (dy)/(dx)=1/(11 square root of x) , y(16)=0
,
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
Apply basic rules of exponents.
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Step 2.3.2.1
Use to rewrite as .
Step 2.3.2.2
Move out of the denominator by raising it to the power.
Step 2.3.2.3
Multiply the exponents in .
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Step 2.3.2.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2.3.2
Combine and .
Step 2.3.2.3.3
Move the negative in front of the fraction.
Step 2.3.3
By the Power Rule, the integral of with respect to is .
Step 2.3.4
Simplify the answer.
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Step 2.3.4.1
Rewrite as .
Step 2.3.4.2
Combine and .
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
Rewrite as .
Step 4.2.2
Apply the power rule and multiply exponents, .
Step 4.2.3
Cancel the common factor of .
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Step 4.2.3.1
Cancel the common factor.
Step 4.2.3.2
Rewrite the expression.
Step 4.2.4
Evaluate the exponent.
Step 4.2.5
Multiply .
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Step 4.2.5.1
Combine and .
Step 4.2.5.2
Multiply by .
Step 4.3
Subtract from both sides of the equation.
Step 5
Substitute for in and simplify.
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Step 5.1
Substitute for .
Step 5.2
Combine and .