Calculus Examples

Solve the Differential Equation (x-y^2)dx+2x(yd)y=0
Step 1
Find where .
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Step 1.1
Differentiate with respect to .
Step 1.2
Differentiate.
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Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.3
Evaluate .
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Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Subtract from .
Step 2
Find where .
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Step 2.1
Differentiate with respect to .
Step 2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Multiply by .
Step 3
Check that .
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Step 3.1
Substitute for and for .
Step 3.2
Since the left side does not equal the right side, the equation is not an identity.
is not an identity.
is not an identity.
Step 4
Find the integration factor .
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Step 4.1
Substitute for .
Step 4.2
Substitute for .
Step 4.3
Substitute for .
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Step 4.3.1
Substitute for .
Step 4.3.2
Simplify the numerator.
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Step 4.3.2.1
Factor out of .
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Step 4.3.2.1.1
Factor out of .
Step 4.3.2.1.2
Factor out of .
Step 4.3.2.1.3
Factor out of .
Step 4.3.2.2
Multiply by .
Step 4.3.2.3
Subtract from .
Step 4.3.3
Cancel the common factor of .
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Step 4.3.3.1
Cancel the common factor.
Step 4.3.3.2
Rewrite the expression.
Step 4.3.4
Cancel the common factor of and .
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Step 4.3.4.1
Factor out of .
Step 4.3.4.2
Cancel the common factors.
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Step 4.3.4.2.1
Factor out of .
Step 4.3.4.2.2
Cancel the common factor.
Step 4.3.4.2.3
Rewrite the expression.
Step 4.3.5
Move the negative in front of the fraction.
Step 4.4
Find the integration factor .
Step 5
Evaluate the integral .
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Step 5.1
Since is constant with respect to , move out of the integral.
Step 5.2
Since is constant with respect to , move out of the integral.
Step 5.3
Multiply by .
Step 5.4
The integral of with respect to is .
Step 5.5
Simplify.
Step 5.6
Simplify each term.
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Step 5.6.1
Simplify by moving inside the logarithm.
Step 5.6.2
Exponentiation and log are inverse functions.
Step 5.6.3
Remove the absolute value in because exponentiations with even powers are always positive.
Step 5.6.4
Rewrite the expression using the negative exponent rule .
Step 6
Multiply both sides of by the integration factor .
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Step 6.1
Multiply by .
Step 6.2
Multiply by .
Step 6.3
Multiply by .
Step 6.4
Cancel the common factor of .
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Step 6.4.1
Factor out of .
Step 6.4.2
Factor out of .
Step 6.4.3
Cancel the common factor.
Step 6.4.4
Rewrite the expression.
Step 6.5
Combine and .
Step 6.6
Combine and .
Step 6.7
Move to the left of .
Step 7
Set equal to the integral of .
Step 8
Integrate to find .
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Step 8.1
Since is constant with respect to , move out of the integral.
Step 8.2
By the Power Rule, the integral of with respect to is .
Step 8.3
Simplify the answer.
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Step 8.3.1
Rewrite as .
Step 8.3.2
Simplify.
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Step 8.3.2.1
Multiply by .
Step 8.3.2.2
Move to the left of .
Step 8.3.2.3
Multiply by .
Step 8.3.2.4
Cancel the common factor of .
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Step 8.3.2.4.1
Cancel the common factor.
Step 8.3.2.4.2
Rewrite the expression.
Step 8.3.2.5
Combine and .
Step 9
Since the integral of will contain an integration constant, we can replace with .
Step 10
Set .
Step 11
Find .
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Step 11.1
Differentiate with respect to .
Step 11.2
By the Sum Rule, the derivative of with respect to is .
Step 11.3
Evaluate .
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Step 11.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 11.3.2
Rewrite as .
Step 11.3.3
Differentiate using the Power Rule which states that is where .
Step 11.4
Differentiate using the function rule which states that the derivative of is .
Step 11.5
Simplify.
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Step 11.5.1
Rewrite the expression using the negative exponent rule .
Step 11.5.2
Combine and .
Step 11.5.3
Reorder terms.
Step 12
Solve for .
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Step 12.1
Solve for .
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Step 12.1.1
Move all terms containing variables to the left side of the equation.
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Step 12.1.1.1
Subtract from both sides of the equation.
Step 12.1.1.2
Combine the numerators over the common denominator.
Step 12.1.1.3
Simplify each term.
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Step 12.1.1.3.1
Apply the distributive property.
Step 12.1.1.3.2
Multiply .
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Step 12.1.1.3.2.1
Multiply by .
Step 12.1.1.3.2.2
Multiply by .
Step 12.1.1.4
Combine the opposite terms in .
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Step 12.1.1.4.1
Add and .
Step 12.1.1.4.2
Add and .
Step 12.1.1.5
Simplify each term.
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Step 12.1.1.5.1
Cancel the common factor of and .
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Step 12.1.1.5.1.1
Factor out of .
Step 12.1.1.5.1.2
Cancel the common factors.
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Step 12.1.1.5.1.2.1
Factor out of .
Step 12.1.1.5.1.2.2
Cancel the common factor.
Step 12.1.1.5.1.2.3
Rewrite the expression.
Step 12.1.1.5.2
Move the negative in front of the fraction.
Step 12.1.2
Add to both sides of the equation.
Step 13
Find the antiderivative of to find .
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Step 13.1
Integrate both sides of .
Step 13.2
Evaluate .
Step 13.3
The integral of with respect to is .
Step 14
Substitute for in .