Solving the Differential Equation ( frac{dy}{dx} frac{y - x}{x} )

When dealing with differential equations, understanding how to approach and solve them is crucial for many applications in mathematics, physics, and engineering. One such equation is:

Problem: Solve the differential equation: ( frac{dy}{dx} frac{y - x}{x} )

Step-by-Step Solution

The given differential equation is:

( frac{dy}{dx} frac{y - x}{x} )

First, we can rewrite the equation to make it more manageable:

( x frac{dy}{dx} y - x )

This form can be further manipulated to highlight the integration on both sides.

Step 1: Rewriting the Equation

By dividing the entire equation by ( x ), we get:

( frac{dy}{dx} frac{y}{x} - 1 )

We can then separate the variables by reorganizing the terms:

( x frac{dy}{dx} x y )

Next, we rearrange the equation to isolate the terms involving ( y ) and ( x ) on different sides:

( x , dy - y , dx -x , dx )

Step 2: Integrating Both Sides

To integrate both sides, we need to perform a process known as integration. The left side of the equation can be integrated by recognizing it as a total differential:

( frac{d}{dx}(xy) y x frac{dy}{dx} )

From the given differential equation ( frac{dy}{dx} frac{y - x}{x} ), we can substitute ( y - frac{y}{x} -1 ) back into the total differential form:

( d(xy) y , dx x , dy )

Therefore, integrating the left side with respect to ( x ):

( xy -x C )

Where ( C ) is an arbitrary constant of integration. Solving this for ( y ):

( y -1 frac{C}{x} )

Step 3: Expressing the Final Answer

The final solution to the differential equation can be written as:

( y -1 frac{C}{x} )

or equivalently:

( y frac{C - x}{x} )

Here, ( C ) is an arbitrary constant, and this form directly incorporates the solution previously derived.

Understanding the Solution

The solution ( y -1 frac{C}{x} ) represents a family of curves on the coordinate plane, each corresponding to a different value of the constant ( C ). This family of solutions is known as the general solution to the differential equation.

Conclusion

Mastering the techniques for solving differential equations, such as those mentioned in the present example, is essential for students and professionals in various fields. By following a step-by-step approach and understanding each transformation, one can solve such equations efficiently.

Keywords: differential equation, separable equation, integration factor, solution method