Solving the Differential Equation y x - y: A Step-by-Step Guide

This article will guide you through the process of solving the differential equation y x - y. The solution involves understanding linear ordinary differential equations and the application of integrating factors. This knowledge is applicable in various fields, including physics, engineering, and mathematics. Let's explore the detailed steps to solve this equation.

Introduction to the Problem

The given differential equation is y x - y. This can be rewritten in the standard form as:

ty y x

Which is a first-order linear ordinary differential equation (ODE). Linear ODEs are widely used in modeling various real-world phenomena, such as population dynamics, radioactive decay, and electrical circuits.

Step-by-Step Solution

Let's solve this differential equation step-by-step:

Step 1: Rewrite the Equation in Standard Form

The given equation is:

ty y x

This can be rewritten as:

ty 1y x

Now, we can identify the coefficients:

ty 1y x

Step 2: Determine the Integrating Factor

The integrating factor (IF) for a first-order linear ODE of the form y Py Q is given by:

text{IF} e^{int Py dx} e^{int 1 dx} e^x

Step 3: Multiply the Entire Equation by the Integrating Factor

Multiplying the entire differential equation by the integrating factor:

te^x(y y) e^x x

This can be rewritten as:

te^x y' e^x y e^x x

The left side is the derivative of the product e^x y with respect to x:

frac{d}{dx}(e^x y) e^x x

Step 4: Integrate Both Sides

Integrating both sides with respect to x gives:

int frac{d}{dx}(e^x y) dx int e^x x dx

The left side is directly the integral of the derivative:

e^x y int e^x x dx

Step 5: Perform Integration by Parts for the Right Side

Performing integration by parts, let:

text{u x, dv e^x dx}

Then, we have:

text{du dx, v e^x}

Using integration by parts:

int e^x x dx x e^x - int e^x dx x e^x - e^x C1

Substituting this back into the equation:

text{e^x y x e^x - e^x C1}

Step 6: Solve for y

Dividing both sides by e^x:

text{y x - 1 C1 e^{-x}}

Since the constant C1 is arbitrary, we can denote it as C for simplicity:

text{y x - 1 - C e^{-x}}

Thus, the general solution of the differential equation y x - y is:

text{y x - 1 - C e^{-x}}

where C is an arbitrary constant.

Alternative Methods

There are alternative methods to solve the same differential equation. Here are two of them:

Method 1: Using the Substitution u x - y

Let u x - y. Then du dy - dx. We need to express dy as follows:

text{dy dx - du}

The given differential equation is:

text{dy x - y}

Substituting u x - y:

text{dx - du x - u}

Rearranging, we get:

text{-du du / (1 - u) dx}

Integrating both sides:

text{-int frac{1}{1 - u} du int dx}

This gives:

text{-ln|1 - u| x C2}

Taking the exponential of both sides:

text{|1 - u| e^{-x - C2}}

Since C2 is arbitrary, we can denote it as C1:

text{1 - u C1 e^{-x}}

Therefore:

text{y x - 1 - C1 e^{-x}}

Method 2: Separation of Variables

Let z x - y. Then, we have:

text{z' 1 - y'}

The differential equation becomes:

text{1 - z' z}

Separating the variables:

text{frac{dz}{1 - z} dx}

Integrating both sides:

text{-ln(1 - z) x C3}

Substituting back z x - y:

text{-ln(1 - (x - y)) x C3}

Taking the exponential of both sides:

text{1 - (x - y) C4 e^{-x}}

Simplifying:

text{y C4 e^{-x} x - 1}

This is essentially the same as the result from the integrating factor method, with the constant term simplified.

Conclusion

In conclusion, the general solution of the differential equation y x - y is:

text{y x - 1 - Ce^{-x}}

where C is an arbitrary constant. This solution can be derived using various methods, such as the integrating factor, substitution, and separation of variables. Understanding these methods is crucial for solving more complex differential equations in the fields of science, engineering, and mathematics.

Related Keywords

differential equation integrating factor linear ordinary differential equation