Solving the Ordinary Differential Equation: x dy - y dx 0

Solving the Ordinary Differential Equation: x dy - y dx 0

In this article, we'll delve into the process of solving the ordinary differential equation x dy - y dx 0. We'll explore various methods and techniques, including separation of variables and integrating factors, to arrive at the general solution. This equation is a first-order differential equation, which can be approached in multiple ways to yield the same result.

Standard Form and Initial Steps

The given differential equation is:

x dy - y dx 0

First, we rewrite the equation in a more standard form:

dy/dx y/x

This form is easier to work with, indicating that we are dealing with a separable differential equation. Separable differential equations can be solved by separating the variables on each side of the equation.

Using Separation of Variables

We express the equation using separation of variables:

dy/y  dx/x

We can now integrate both sides of the equation:

∫ (1/y) dy  ∫ (1/x) dx

The integration yields:

ln|y|  ln|x|   C

Exponentiating both sides to solve for y:

|y|  e^(ln|x|   C)  e^C * |x|

Since e^C is just another constant, we denote it as C (or K for simplicity), thus:

y  K * x

This is the general solution to the differential equation, where K is an arbitrary constant.

Alternative Method: Substitution

Let's explore another method using substitution to solve the given differential equation. We start with the equation:

x dy - y dx 0

We can rewrite it as:

x (dy/dx) y

Or equivalently:

dy/dx y/x

We can use the substitution method by setting:

v x/y

Then, we differentiate v with respect to x using the product rule:

dv/dx (x/y)'

Which simplifies to:

dv/dx (1/y) - (x/y^2) (dy/dx)

Substituting dy/dx y/x into the equation:

dv/dx (1/y) - (x/y^2) (y/x)

Simplifying further:

dv/dx (1/y) - (1/y)

This simplifies to:

dv/dx 0

Which implies:

v C

Substituting back v x/y to get:

x/y C

Thus, the solution is:

y x/C K * x

Again, this confirms the general solution of the differential equation.

Conclusion

In conclusion, we have successfully solved the ordinary differential equation x dy - y dx 0 using both separation of variables and substitution methods. The general solution is:

y Kx

Where K is an arbitrary constant. This solution represents the family of curves that satisfy the given differential equation.