Calculus Examples

Solve the Differential Equation
,
Separate the variables.
Tap for more steps...
Divide each term in by and simplify.
Combine the numerators over the common denominator.
Multiply both sides by .
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Rewrite the expression.
Rewrite the equation.
Integrate both sides.
Tap for more steps...
Set up an integral on each side.
Integrate the left side.
Tap for more steps...
Let . Then , so . Rewrite using and .
Tap for more steps...
Let . Find .
Tap for more steps...
Differentiate .
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Tap for more steps...
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 .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Rewrite the problem using and .
Simplify.
Tap for more steps...
Multiply by .
Move to the left of .
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Simplify.
Replace all occurrences of with .
The integral of with respect to is .
Group the constant of integration on the right side as .
Solve for .
Tap for more steps...
Multiply both sides of the equation by .
Simplify both sides of the equation.
Tap for more steps...
Simplify the left side.
Tap for more steps...
Simplify .
Tap for more steps...
Combine and .
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Rewrite the expression.
Simplify the right side.
Tap for more steps...
Apply the distributive property.
Move all the terms containing a logarithm to the left side of the equation.
Simplify the left side.
Tap for more steps...
Simplify .
Tap for more steps...
Simplify each term.
Tap for more steps...
Simplify by moving inside the logarithm.
Remove the absolute value in because exponentiations with even powers are always positive.
Use the quotient property of logarithms, .
To solve for , rewrite the equation using properties of logarithms.
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Solve for .
Tap for more steps...
Rewrite the equation as .
Multiply both sides by .
Simplify.
Tap for more steps...
Simplify the left side.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Rewrite the expression.
Simplify the right side.
Tap for more steps...
Reorder factors in .
Solve for .
Tap for more steps...
Remove the absolute value term. This creates a on the right side of the equation because .
Add to both sides of the equation.
Divide each term in by and simplify.
Tap for more steps...
Divide each term in by .
Simplify the left side.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Divide by .
Group the constant terms together.
Tap for more steps...
Simplify the constant of integration.
Combine constants with the plus or minus.
Use the initial condition to find the value of by substituting for and for in .
Solve for .
Tap for more steps...
Rewrite the equation as .
Simplify each term.
Tap for more steps...
One to any power is one.
Multiply by .
Move all terms not containing to the right side of the equation.
Tap for more steps...
Subtract from both sides of the equation.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Subtract from .
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Substitute for in and simplify.
Tap for more steps...
Substitute.
Multiply by .
MathMaster requires javascript and a modern browser.