Calculus Examples

Solve the Differential Equation x^2(dy)/(dx)=xy+x^2+y^2
Step 1
Rewrite the differential equation as a function of .
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Step 1.1
Divide each term in by and simplify.
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Step 1.1.1
Divide each term in by .
Step 1.1.2
Simplify the left side.
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Step 1.1.2.1
Cancel the common factor of .
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Step 1.1.2.1.1
Cancel the common factor.
Step 1.1.2.1.2
Divide by .
Step 1.1.3
Simplify the right side.
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Step 1.1.3.1
Simplify each term.
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Step 1.1.3.1.1
Cancel the common factor of and .
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Step 1.1.3.1.1.1
Factor out of .
Step 1.1.3.1.1.2
Cancel the common factors.
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Step 1.1.3.1.1.2.1
Factor out of .
Step 1.1.3.1.1.2.2
Cancel the common factor.
Step 1.1.3.1.1.2.3
Rewrite the expression.
Step 1.1.3.1.2
Cancel the common factor of .
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Step 1.1.3.1.2.1
Cancel the common factor.
Step 1.1.3.1.2.2
Rewrite the expression.
Step 1.2
Rewrite as .
Step 2
Let . Substitute for .
Step 3
Solve for .
Step 4
Use the product rule to find the derivative of with respect to .
Step 5
Substitute for .
Step 6
Solve the substituted differential equation.
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Step 6.1
Separate the variables.
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Step 6.1.1
Solve for .
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Step 6.1.1.1
Move all terms not containing to the right side of the equation.
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Step 6.1.1.1.1
Subtract from both sides of the equation.
Step 6.1.1.1.2
Combine the opposite terms in .
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Step 6.1.1.1.2.1
Subtract from .
Step 6.1.1.1.2.2
Add and .
Step 6.1.1.2
Divide each term in by and simplify.
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Step 6.1.1.2.1
Divide each term in by .
Step 6.1.1.2.2
Simplify the left side.
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Step 6.1.1.2.2.1
Cancel the common factor of .
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Step 6.1.1.2.2.1.1
Cancel the common factor.
Step 6.1.1.2.2.1.2
Divide by .
Step 6.1.2
Combine the numerators over the common denominator.
Step 6.1.3
Multiply both sides by .
Step 6.1.4
Cancel the common factor of .
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Step 6.1.4.1
Cancel the common factor.
Step 6.1.4.2
Rewrite the expression.
Step 6.1.5
Rewrite the equation.
Step 6.2
Integrate both sides.
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Step 6.2.1
Set up an integral on each side.
Step 6.2.2
Integrate the left side.
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Step 6.2.2.1
Rewrite as .
Step 6.2.2.2
The integral of with respect to is .
Step 6.2.3
The integral of with respect to is .
Step 6.2.4
Group the constant of integration on the right side as .
Step 6.3
Take the inverse arctangent of both sides of the equation to extract from inside the arctangent.
Step 7
Substitute for .
Step 8
Solve for .
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Step 8.1
Multiply both sides by .
Step 8.2
Simplify.
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Step 8.2.1
Simplify the left side.
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Step 8.2.1.1
Cancel the common factor of .
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Step 8.2.1.1.1
Cancel the common factor.
Step 8.2.1.1.2
Rewrite the expression.
Step 8.2.2
Simplify the right side.
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Step 8.2.2.1
Reorder factors in .