# Calculus Examples

Solve the Differential Equation
Step 1
To solve the differential equation, let where is the exponent of .
Step 2
Solve the equation for .
Step 3
Take the derivative of with respect to .
Step 4
Take the derivative of with respect to .
Take the derivative of .
Rewrite the expression using the negative exponent rule .
Differentiate using the Quotient Rule which states that is where and .
Differentiate using the Constant Rule.
Multiply by .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Rewrite as .
Step 5
Substitute for in the original equation .
Step 6
Substitute for in the original equation .
Step 7
Solve the substituted differential equation.
Substitute for in the original equation .
Substitute for in the original equation .
Multiply each term in by to eliminate the fractions.
Multiply each term in by .
Simplify the left side.
Simplify each term.
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Subtract from .
Simplify .
Multiply by .
Multiply by .
Simplify the right side.
Rewrite using the commutative property of multiplication.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Subtract from .
Simplify .
The integrating factor is defined by the formula , where .
Set up the integration.
Apply the constant rule.
Remove the constant of integration.
Multiply each term by the integrating factor .
Multiply each term by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Reorder factors in .
Rewrite the left side as a result of differentiating a product.
Set up an integral on each side.
Integrate the left side.
Integrate the right side.
Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
Let . Find .
Differentiate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Rewrite the problem using and .
Combine and .
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Simplify.
Replace all occurrences of with .
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Simplify each term.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Multiply by .
Cancel the common factor.
Rewrite the expression.
Divide by .
Combine and .
Step 8
Substitute for .