Calculus Examples

Solve the Differential Equation 2x(dy)/(dx)-y=0
Step 1
Separate the variables.
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Step 1.1
Solve for .
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Step 1.1.1
Add to both sides of the equation.
Step 1.1.2
Divide each term in by and simplify.
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Step 1.1.2.1
Divide each term in by .
Step 1.1.2.2
Simplify the left side.
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Step 1.1.2.2.1
Cancel the common factor of .
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Step 1.1.2.2.1.1
Cancel the common factor.
Step 1.1.2.2.1.2
Rewrite the expression.
Step 1.1.2.2.2
Cancel the common factor of .
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Step 1.1.2.2.2.1
Cancel the common factor.
Step 1.1.2.2.2.2
Divide by .
Step 1.2
Multiply both sides by .
Step 1.3
Cancel the common factor of .
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Step 1.3.1
Cancel the common factor.
Step 1.3.2
Rewrite the expression.
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
The integral of with respect to is .
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
The integral of with respect to is .
Step 2.3.3
Simplify.
Step 2.4
Group the constant of integration on the right side as .
Step 3
Solve for .
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Step 3.1
Simplify the right side.
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Step 3.1.1
Combine and .
Step 3.2
Move all the terms containing a logarithm to the left side of the equation.
Step 3.3
Simplify the left side.
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Step 3.3.1
Simplify .
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Step 3.3.1.1
Simplify each term.
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Step 3.3.1.1.1
Rewrite as .
Step 3.3.1.1.2
Simplify by moving inside the logarithm.
Step 3.3.1.2
Use the quotient property of logarithms, .
Step 3.4
To solve for , rewrite the equation using properties of logarithms.
Step 3.5
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 3.6
Solve for .
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Step 3.6.1
Rewrite the equation as .
Step 3.6.2
Multiply both sides by .
Step 3.6.3
Simplify the left side.
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Step 3.6.3.1
Cancel the common factor of .
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Step 3.6.3.1.1
Cancel the common factor.
Step 3.6.3.1.2
Rewrite the expression.
Step 3.6.4
Remove the absolute value term. This creates a on the right side of the equation because .
Step 4
Group the constant terms together.
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Step 4.1
Simplify the constant of integration.
Step 4.2
Combine constants with the plus or minus.