Enter a problem...
Calculus Examples
Step 1
Let . Substitute for all occurrences of .
Step 2
Step 2.1
Differentiate using the chain rule, which states that is where and .
Step 2.1.1
To apply the Chain Rule, set as .
Step 2.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.1.3
Replace all occurrences of with .
Step 2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
Rewrite as .
Step 3
Substitute for .
Step 4
Substitute the derivative back in to the differential equation.
Step 5
Step 5.1
Solve for .
Step 5.1.1
Subtract from both sides of the equation.
Step 5.1.2
Divide each term in by and simplify.
Step 5.1.2.1
Divide each term in by .
Step 5.1.2.2
Simplify the left side.
Step 5.1.2.2.1
Dividing two negative values results in a positive value.
Step 5.1.2.2.2
Divide by .
Step 5.1.2.3
Simplify the right side.
Step 5.1.2.3.1
Simplify each term.
Step 5.1.2.3.1.1
Move the negative one from the denominator of .
Step 5.1.2.3.1.2
Rewrite as .
Step 5.1.2.3.1.3
Multiply by .
Step 5.1.2.3.1.4
Divide by .
Step 5.1.3
Multiply both sides by .
Step 5.1.4
Simplify.
Step 5.1.4.1
Simplify the left side.
Step 5.1.4.1.1
Cancel the common factor of .
Step 5.1.4.1.1.1
Cancel the common factor.
Step 5.1.4.1.1.2
Rewrite the expression.
Step 5.1.4.2
Simplify the right side.
Step 5.1.4.2.1
Simplify .
Step 5.1.4.2.1.1
Apply the distributive property.
Step 5.1.4.2.1.2
Multiply by by adding the exponents.
Step 5.1.4.2.1.2.1
Move .
Step 5.1.4.2.1.2.2
Multiply by .
Step 5.1.4.2.1.3
Multiply by .
Step 5.2
Multiply both sides by .
Step 5.3
Simplify.
Step 5.3.1
Factor out of .
Step 5.3.1.1
Factor out of .
Step 5.3.1.2
Raise to the power of .
Step 5.3.1.3
Factor out of .
Step 5.3.1.4
Factor out of .
Step 5.3.2
Multiply by .
Step 5.3.3
Factor out of .
Step 5.3.3.1
Factor out of .
Step 5.3.3.2
Raise to the power of .
Step 5.3.3.3
Factor out of .
Step 5.3.3.4
Factor out of .
Step 5.3.4
Cancel the common factor of .
Step 5.3.4.1
Cancel the common factor.
Step 5.3.4.2
Rewrite the expression.
Step 5.3.5
Cancel the common factor of .
Step 5.3.5.1
Cancel the common factor.
Step 5.3.5.2
Rewrite the expression.
Step 5.4
Rewrite the equation.
Step 6
Step 6.1
Set up an integral on each side.
Step 6.2
Integrate the left side.
Step 6.2.1
Write the fraction using partial fraction decomposition.
Step 6.2.1.1
Decompose the fraction and multiply through by the common denominator.
Step 6.2.1.1.1
Factor out of .
Step 6.2.1.1.1.1
Factor out of .
Step 6.2.1.1.1.2
Raise to the power of .
Step 6.2.1.1.1.3
Factor out of .
Step 6.2.1.1.1.4
Factor out of .
Step 6.2.1.1.2
For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .
Step 6.2.1.1.3
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 6.2.1.1.4
Cancel the common factor of .
Step 6.2.1.1.4.1
Cancel the common factor.
Step 6.2.1.1.4.2
Rewrite the expression.
Step 6.2.1.1.5
Cancel the common factor of .
Step 6.2.1.1.5.1
Cancel the common factor.
Step 6.2.1.1.5.2
Rewrite the expression.
Step 6.2.1.1.6
Simplify each term.
Step 6.2.1.1.6.1
Cancel the common factor of .
Step 6.2.1.1.6.1.1
Cancel the common factor.
Step 6.2.1.1.6.1.2
Divide by .
Step 6.2.1.1.6.2
Apply the distributive property.
Step 6.2.1.1.6.3
Rewrite using the commutative property of multiplication.
Step 6.2.1.1.6.4
Multiply by .
Step 6.2.1.1.6.5
Cancel the common factor of .
Step 6.2.1.1.6.5.1
Cancel the common factor.
Step 6.2.1.1.6.5.2
Divide by .
Step 6.2.1.1.7
Simplify the expression.
Step 6.2.1.1.7.1
Move .
Step 6.2.1.1.7.2
Reorder and .
Step 6.2.1.1.7.3
Move .
Step 6.2.1.2
Create equations for the partial fraction variables and use them to set up a system of equations.
Step 6.2.1.2.1
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 6.2.1.2.2
Create an equation for the partial fraction variables by equating the coefficients of the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 6.2.1.2.3
Set up the system of equations to find the coefficients of the partial fractions.
Step 6.2.1.3
Solve the system of equations.
Step 6.2.1.3.1
Rewrite the equation as .
Step 6.2.1.3.2
Replace all occurrences of with in each equation.
Step 6.2.1.3.2.1
Replace all occurrences of in with .
Step 6.2.1.3.2.2
Simplify the right side.
Step 6.2.1.3.2.2.1
Multiply by .
Step 6.2.1.3.3
Solve for in .
Step 6.2.1.3.3.1
Rewrite the equation as .
Step 6.2.1.3.3.2
Add to both sides of the equation.
Step 6.2.1.3.4
Solve the system of equations.
Step 6.2.1.3.5
List all of the solutions.
Step 6.2.1.4
Replace each of the partial fraction coefficients in with the values found for and .
Step 6.2.1.5
Simplify.
Step 6.2.1.5.1
Factor out of .
Step 6.2.1.5.2
Rewrite as .
Step 6.2.1.5.3
Factor out of .
Step 6.2.1.5.4
Rewrite negatives.
Step 6.2.1.5.4.1
Rewrite as .
Step 6.2.1.5.4.2
Move the negative in front of the fraction.
Step 6.2.2
Split the single integral into multiple integrals.
Step 6.2.3
The integral of with respect to is .
Step 6.2.4
Since is constant with respect to , move out of the integral.
Step 6.2.5
Since is constant with respect to , move out of the integral.
Step 6.2.6
Multiply by .
Step 6.2.7
Let . Then , so . Rewrite using and .
Step 6.2.7.1
Let . Find .
Step 6.2.7.1.1
Differentiate .
Step 6.2.7.1.2
By the Sum Rule, the derivative of with respect to is .
Step 6.2.7.1.3
Evaluate .
Step 6.2.7.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.2.7.1.3.2
Differentiate using the Power Rule which states that is where .
Step 6.2.7.1.3.3
Multiply by .
Step 6.2.7.1.4
Differentiate using the Constant Rule.
Step 6.2.7.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.2.7.1.4.2
Add and .
Step 6.2.7.2
Rewrite the problem using and .
Step 6.2.8
Simplify.
Step 6.2.8.1
Multiply by .
Step 6.2.8.2
Move to the left of .
Step 6.2.9
Since is constant with respect to , move out of the integral.
Step 6.2.10
Simplify.
Step 6.2.10.1
Combine and .
Step 6.2.10.2
Cancel the common factor of and .
Step 6.2.10.2.1
Factor out of .
Step 6.2.10.2.2
Cancel the common factors.
Step 6.2.10.2.2.1
Factor out of .
Step 6.2.10.2.2.2
Cancel the common factor.
Step 6.2.10.2.2.3
Rewrite the expression.
Step 6.2.10.2.2.4
Divide by .
Step 6.2.11
The integral of with respect to is .
Step 6.2.12
Simplify.
Step 6.3
Apply the constant rule.
Step 6.4
Group the constant of integration on the right side as .
Step 7
Step 7.1
Use the quotient property of logarithms, .
Step 7.2
Reorder and .
Step 7.3
To solve for , rewrite the equation using properties of logarithms.
Step 7.4
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 7.5
Solve for .
Step 7.5.1
Rewrite the equation as .
Step 7.5.2
Multiply both sides by .
Step 7.5.3
Simplify the left side.
Step 7.5.3.1
Cancel the common factor of .
Step 7.5.3.1.1
Cancel the common factor.
Step 7.5.3.1.2
Rewrite the expression.
Step 7.5.4
Solve for .
Step 7.5.4.1
Reorder factors in .
Step 7.5.4.2
Remove the absolute value term. This creates a on the right side of the equation because .
Step 8
Step 8.1
Reorder terms.
Step 8.2
Rewrite as .
Step 8.3
Reorder and .
Step 8.4
Combine constants with the plus or minus.
Step 9
Replace all occurrences of with .
Step 10
Step 10.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 10.2
Expand the left side.
Step 10.2.1
Expand by moving outside the logarithm.
Step 10.2.2
The natural logarithm of is .
Step 10.2.3
Multiply by .
Step 10.3
Expand the right side.
Step 10.3.1
Rewrite as .
Step 10.3.2
Rewrite as .
Step 10.3.3
Expand by moving outside the logarithm.
Step 10.3.4
The natural logarithm of is .
Step 10.3.5
Multiply by .
Step 10.4
Simplify the right side.
Step 10.4.1
Use the product property of logarithms, .
Step 10.5
Move all terms not containing to the right side of the equation.
Step 10.5.1
Subtract from both sides of the equation.
Step 10.5.2
Combine the opposite terms in .
Step 10.5.2.1
Subtract from .
Step 10.5.2.2
Add and .
Step 10.6
Divide each term in by and simplify.
Step 10.6.1
Divide each term in by .
Step 10.6.2
Simplify the left side.
Step 10.6.2.1
Dividing two negative values results in a positive value.
Step 10.6.2.2
Divide by .
Step 10.6.3
Simplify the right side.
Step 10.6.3.1
Move the negative one from the denominator of .
Step 10.6.3.2
Rewrite as .