# Solve the Differential Equation (dy)/(dx)=xy-4x

Separate the variables.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Multiply both sides by .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite the equation.
Integrate both sides.
Set up an integral on each side.
Integrate the left side.
Let . Then . Rewrite using and .
Let . Find .
Differentiate .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Rewrite the problem using and .
The integral of with respect to is .
Replace all occurrences of with .
By the Power Rule, the integral of with respect to is .
Group the constant of integration on the right side as .
Solve for .
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 .
Rewrite the equation as .
Combine and .
Remove the absolute value term. This creates a on the right side of the equation because .
Add to both sides of the equation.
Group the constant terms together.
Rewrite as .
Reorder and .
Combine constants with the plus or minus.
Solve the Differential Equation (dy)/(dx)=xy-4x