Calculus Examples

Solve the Differential Equation (dy)/(dx)=2x(e^(-y))
Step 1
Separate the variables.
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Step 1.1
Multiply both sides by .
Step 1.2
Simplify.
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Step 1.2.1
Rewrite using the commutative property of multiplication.
Step 1.2.2
Combine and .
Step 1.2.3
Cancel the common factor of .
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Step 1.2.3.1
Factor out of .
Step 1.2.3.2
Cancel the common factor.
Step 1.2.3.3
Rewrite the expression.
Step 1.3
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
Integrate the left side.
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Step 2.2.1
Simplify the expression.
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Step 2.2.1.1
Negate the exponent of and move it out of the denominator.
Step 2.2.1.2
Simplify.
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Step 2.2.1.2.1
Multiply the exponents in .
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Step 2.2.1.2.1.1
Apply the power rule and multiply exponents, .
Step 2.2.1.2.1.2
Multiply .
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Step 2.2.1.2.1.2.1
Multiply by .
Step 2.2.1.2.1.2.2
Multiply by .
Step 2.2.1.2.2
Multiply by .
Step 2.2.2
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
By the Power Rule, the integral of with respect to is .
Step 2.3.3
Simplify the answer.
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Step 2.3.3.1
Rewrite as .
Step 2.3.3.2
Simplify.
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Step 2.3.3.2.1
Combine and .
Step 2.3.3.2.2
Cancel the common factor of .
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Step 2.3.3.2.2.1
Cancel the common factor.
Step 2.3.3.2.2.2
Rewrite the expression.
Step 2.3.3.2.3
Multiply by .
Step 2.4
Group the constant of integration on the right side as .
Step 3
Solve for .
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Step 3.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.2
Expand the left side.
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Step 3.2.1
Expand by moving outside the logarithm.
Step 3.2.2
The natural logarithm of is .
Step 3.2.3
Multiply by .