(x − Y) Dx + X Dy = 0

3 min read Jun 16, 2024
(x − Y) Dx + X Dy = 0

Solving the Differential Equation: (x − y) dx + x dy = 0

This article will delve into the solution of the differential equation (x − y) dx + x dy = 0. We'll explore various techniques and arrive at a general solution.

Recognizing the Type of Differential Equation

First, let's analyze the given equation to determine its type. We can see that the equation is a first-order homogeneous differential equation. This means it can be expressed in the form:

dy/dx = f(x,y)

where f(x, y) is a function of x and y that is homogeneous of degree zero. In other words, f(tx, ty) = f(x, y) for any non-zero value of t.

Solving by Substitution

To solve this equation, we can use the substitution y = vx. This implies that:

dy/dx = v + x dv/dx

Substituting these expressions into the original differential equation, we get:

(x - vx) dx + x(v + x dv/dx) = 0

Simplifying, we obtain:

x dx + x² dv/dx = 0

Dividing both sides by x² and rearranging, we get:

dv/dx = -1/x

Integration and Finding the General Solution

Now, we can integrate both sides with respect to x:

∫ dv = -∫ (1/x) dx

This leads to:

v = -ln|x| + C

where C is the constant of integration.

Finally, substituting back v = y/x, we get the general solution:

y/x = -ln|x| + C

Rearranging, we obtain the final solution:

y = -x ln|x| + Cx

Conclusion

We have successfully solved the differential equation (x − y) dx + x dy = 0 using the method of substitution. The solution is y = -x ln|x| + Cx, where C is an arbitrary constant. This solution represents a family of curves that are solutions to the given differential equation.