# Calculus Examples

Solve the Differential Equation
,
Step 1
Rewrite the equation.
Step 2
Integrate both sides.
Step 2.1
Set up an integral on each side.
Step 2.2
Apply the constant rule.
Step 2.3
Integrate the right side.
Step 2.3.1
Let . Then , so . Rewrite using and .
Step 2.3.1.1
Let . Find .
Step 2.3.1.1.1
Differentiate .
Step 2.3.1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.1.1.3
Differentiate using the Power Rule which states that is where .
Step 2.3.1.1.4
Multiply by .
Step 2.3.1.2
Rewrite the problem using and .
Step 2.3.2
Simplify.
Step 2.3.2.1
Move the negative in front of the fraction.
Step 2.3.2.2
Combine and .
Step 2.3.3
Since is constant with respect to , move out of the integral.
Step 2.3.4
Since is constant with respect to , move out of the integral.
Step 2.3.5
The integral of with respect to is .
Step 2.3.6
Simplify.
Step 2.3.7
Replace all occurrences of with .
Step 2.4
Group the constant of integration on the right side as .
Step 3
Use the initial condition to find the value of by substituting for and for in .
Step 4
Solve for .
Step 4.1
Rewrite the equation as .
Step 4.2
Simplify each term.
Step 4.2.1
Multiply by .
Step 4.2.2
Anything raised to is .
Step 4.2.3
Multiply by .
Step 4.3
Move all terms not containing to the right side of the equation.
Step 4.3.1
Add to both sides of the equation.
Step 4.3.2
To write as a fraction with a common denominator, multiply by .
Step 4.3.3
Combine and .
Step 4.3.4
Combine the numerators over the common denominator.
Step 4.3.5
Simplify the numerator.
Step 4.3.5.1
Multiply by .
Step 4.3.5.2