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

When dealing with a differential equation like (frac{dy}{dx} frac{y - x}{x y}), it is essential to recognize that the equation is (f(tx, ty) t^0 f(x, y)), meaning it is homogeneous. To solve such equations, we can use a substitution method based on this property. Let's explore the step-by-step process to find the solution.

Step 1: Substitution

The substitution we will use is (v frac{y}{x}). This allows us to express (y) in terms of (x) and (v), specifically (y vx).

Using the substitution (y vx), we can differentiate (y) with respect to (x) and get:

(frac{dy}{dx} v frac{dx}{dx} x frac{dv}{dx} v x frac{dv}{dx})

Step 2: Substitute into the Equation

Now, we substitute (y vx) and (frac{dy}{dx} v x frac{dv}{dx}) into the original equation:

(v x frac{dv}{dx} frac{vx - x}{x vx})

Step 3: Simplify the Right-Hand Side

The right-hand side of the equation simplifies as follows:

(frac{vx - x}{x vx} frac{x(v - 1)}{x(1 v)} frac{v - 1}{1 v})

Thus, the equation becomes:

(v x frac{dv}{dx} frac{v - 1}{1 v})

Step 4: Rearrange the Equation

Rearranging the equation, we isolate (x frac{dv}{dx}) on one side:

(x frac{dv}{dx} frac{v - 1}{1 v} - v)

Combining the terms on the right-hand side:

(x frac{dv}{dx} frac{v - 1 - v(1 v)}{1 v} frac{v - 1 - v - v^2}{1 v} frac{-1 - v^2}{1 v})

Step 5: Separate Variables

Separting the variables, we get:

(frac{1 v}{-1 - v^2} dv frac{dx}{x})

Step 6: Integrate Both Sides

Integrating both sides, we start by simplifying the left side:

(int frac{1 v}{-1 - v^2} dv int frac{-1 - v}{1 v^2} dv)

This can be split into two integrals:

(-int frac{1}{1 v^2} dv - int frac{v}{1 v^2} dv)

The first integral gives:

(-tan^{-1} v)

The second integral can be solved using the substitution (u 1 v^2), which gives:

(-frac{1}{2}ln(1 v^2))

Thus, we have:

(-tan^{-1} v - frac{1}{2}ln(1 v^2) ln x C)

Step 7: Back-Substitute (v frac{y}{x})

Finally, we back-substitute (v frac{y}{x}) into the solution:

(-tan^{-1}left(frac{y}{x}right) - frac{1}{2}lnleft(1 left(frac{y}{x}right)^2right) ln x C)

This implicit solution can be further manipulated or solved for (y) if needed.

Conclusion

By following the steps outlined above, we have successfully solved the homogeneous differential equation (frac{dy}{dx} frac{y - x}{x y}). Understanding this method can help in tackling similar differential equations efficiently.