Calculus Examples

Solve the Differential Equation (1+y)dx=(1-x)dy
Step 1
Rewrite the equation.
Step 2
Multiply both sides by .
Step 3
Simplify.
Tap for more steps...
Step 3.1
Cancel the common factor of .
Tap for more steps...
Step 3.1.1
Cancel the common factor.
Step 3.1.2
Rewrite the expression.
Step 3.2
Cancel the common factor of .
Tap for more steps...
Step 3.2.1
Factor out of .
Step 3.2.2
Cancel the common factor.
Step 3.2.3
Rewrite the expression.
Step 4
Integrate both sides.
Tap for more steps...
Step 4.1
Set up an integral on each side.
Step 4.2
Integrate the left side.
Tap for more steps...
Step 4.2.1
Let . Then . Rewrite using and .
Tap for more steps...
Step 4.2.1.1
Let . Find .
Tap for more steps...
Step 4.2.1.1.1
Differentiate .
Step 4.2.1.1.2
By the Sum Rule, the derivative of with respect to is .
Step 4.2.1.1.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.2.1.1.4
Differentiate using the Power Rule which states that is where .
Step 4.2.1.1.5
Add and .
Step 4.2.1.2
Rewrite the problem using and .
Step 4.2.2
The integral of with respect to is .
Step 4.2.3
Replace all occurrences of with .
Step 4.3
Integrate the right side.
Tap for more steps...
Step 4.3.1
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 4.3.1.1
Let . Find .
Tap for more steps...
Step 4.3.1.1.1
Rewrite.
Step 4.3.1.1.2
Divide by .
Step 4.3.1.2
Rewrite the problem using and .
Step 4.3.2
Split the fraction into multiple fractions.
Step 4.3.3
Since is constant with respect to , move out of the integral.
Step 4.3.4
The integral of with respect to is .
Step 4.3.5
Simplify.
Step 4.3.6
Replace all occurrences of with .
Step 4.4
Group the constant of integration on the right side as .
Step 5
Solve for .
Tap for more steps...
Step 5.1
Move all the terms containing a logarithm to the left side of the equation.
Step 5.2
Use the product property of logarithms, .
Step 5.3
To multiply absolute values, multiply the terms inside each absolute value.
Step 5.4
Expand using the FOIL Method.
Tap for more steps...
Step 5.4.1
Apply the distributive property.
Step 5.4.2
Apply the distributive property.
Step 5.4.3
Apply the distributive property.
Step 5.5
Simplify each term.
Tap for more steps...
Step 5.5.1
Multiply by .
Step 5.5.2
Multiply by .
Step 5.5.3
Multiply by .
Step 5.5.4
Rewrite using the commutative property of multiplication.
Step 5.6
To solve for , rewrite the equation using properties of logarithms.
Step 5.7
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 5.8
Solve for .
Tap for more steps...
Step 5.8.1
Rewrite the equation as .
Step 5.8.2
Remove the absolute value term. This creates a on the right side of the equation because .
Step 5.8.3
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 5.8.3.1
Subtract from both sides of the equation.
Step 5.8.3.2
Add to both sides of the equation.
Step 5.8.4
Factor out of .
Tap for more steps...
Step 5.8.4.1
Factor out of .
Step 5.8.4.2
Factor out of .
Step 5.8.4.3
Factor out of .
Step 5.8.5
Rewrite as .
Step 5.8.6
Divide each term in by and simplify.
Tap for more steps...
Step 5.8.6.1
Divide each term in by .
Step 5.8.6.2
Simplify the left side.
Tap for more steps...
Step 5.8.6.2.1
Cancel the common factor of .
Tap for more steps...
Step 5.8.6.2.1.1
Cancel the common factor.
Step 5.8.6.2.1.2
Divide by .
Step 5.8.6.3
Simplify the right side.
Tap for more steps...
Step 5.8.6.3.1
Move the negative in front of the fraction.
Step 5.8.6.3.2
Combine the numerators over the common denominator.
Step 5.8.6.3.3
Combine the numerators over the common denominator.
Step 6
Simplify the constant of integration.