-dx-%2B-x-dy-%3D-0.jpeg "(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.