# Calculus Examples

Solve the Differential Equation
Step 1
Rewrite the differential equation as a function of .
Step 1.1
Assume .
Step 1.2
Combine and into a single radical.
Step 1.3
Reduce the expression by cancelling the common factors.
Step 1.3.1
Factor out of .
Step 1.3.2
Factor out of .
Step 1.3.3
Cancel the common factor.
Step 1.3.4
Rewrite the expression.
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.
Step 6.1
Separate the variables.
Step 6.1.1
Solve for .
Step 6.1.1.1
Move all terms not containing to the right side of the equation.
Step 6.1.1.1.1
Subtract from both sides of the equation.
Step 6.1.1.1.2
Combine the opposite terms in .
Step 6.1.1.1.2.1
Subtract from .
Step 6.1.1.1.2.2
Step 6.1.1.2
Divide each term in by and simplify.
Step 6.1.1.2.1
Divide each term in by .
Step 6.1.1.2.2
Simplify the left side.
Step 6.1.1.2.2.1
Cancel the common factor of .
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
Multiply both sides by .
Step 6.1.3
Cancel the common factor of .
Step 6.1.3.1
Cancel the common factor.
Step 6.1.3.2
Rewrite the expression.
Step 6.1.4
Rewrite the equation.
Step 6.2
Integrate both sides.
Step 6.2.1
Set up an integral on each side.
Step 6.2.2
Integrate the left side.
Step 6.2.2.1
Apply basic rules of exponents.
Step 6.2.2.1.1
Use to rewrite as .
Step 6.2.2.1.2
Move out of the denominator by raising it to the power.
Step 6.2.2.1.3
Multiply the exponents in .
Step 6.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 6.2.2.1.3.2
Combine and .
Step 6.2.2.1.3.3
Move the negative in front of the fraction.
Step 6.2.2.2
By the Power Rule, 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
Solve for .
Step 6.3.1
Divide each term in by and simplify.
Step 6.3.1.1
Divide each term in by .
Step 6.3.1.2
Simplify the left side.
Step 6.3.1.2.1
Cancel the common factor.
Step 6.3.1.2.2
Divide by .
Step 6.3.1.3
Simplify the right side.
Step 6.3.1.3.1
Simplify each term.
Step 6.3.1.3.1.1
Rewrite as .
Step 6.3.1.3.1.2
Simplify by moving inside the logarithm.
Step 6.3.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 6.3.3
Simplify the left side.
Step 6.3.3.1
Simplify .
Step 6.3.3.1.1
Multiply the exponents in .
Step 6.3.3.1.1.1
Apply the power rule and multiply exponents, .
Step 6.3.3.1.1.2
Cancel the common factor of .
Step 6.3.3.1.1.2.1
Cancel the common factor.
Step 6.3.3.1.1.2.2
Rewrite the expression.
Step 6.3.3.1.2
Simplify.
Step 6.4
Simplify the constant of integration.
Step 7
Substitute for .
Step 8
Solve for .
Step 8.1
Multiply both sides by .
Step 8.2
Simplify.
Step 8.2.1
Simplify the left side.
Step 8.2.1.1
Cancel the common factor of .
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.
Step 8.2.2.1
Reorder factors in .