Solving the Differential Equation ( ln y frac{dy}{dx} -x ): Analyzing Multiple Methods and Their Solutions
This article explores the solution of the differential equation ( ln y frac{dy}{dx} -x ) through multiple analytical approaches. We will discuss the process, various interpretations, and derive the general solution for each method.
Introduction to the Differential Equation
The given differential equation is ( ln y frac{dy}{dx} -x ). This is a separable differential equation, allowing us to separate the variables for easier integration.
Method 1: Variables Separable Form
Step-by-Step Solution
Starting from the equation:
[ ln y frac{dy}{dx} -x ]
We can rewrite it as:
[ ln y dy -x dx ]
Integrating both sides:
[ int ln y dy -int x dx C_1 ]
Using integration by parts for the left side, where ( u ln y ) and ( dv dy ), we get:
[ y ln y - int frac{y}{y} dy -frac{x^2}{2} C ]
Simplifying:
[ y ln y - y -frac{x^2}{2} C ]
This is the general solution for the first method.
Method 2: Another Interpretation
Alternative Solution
Consider the second interpretation:
[ ln y frac{dy}{dx} -x ]
This can be separated into:
[ ln y dy -x dx ]
Integrating both sides:
[ int ln y dy -int x dx ]
Using integration by parts for the left side, where ( u ln y ) and ( dv dy ), we get:
[ y ln y - y -frac{x^2}{2} C ]
This confirms the same general solution as before.
Another Interpretation of the Integral
Exploring a Different Path
Another way to interpret the integral is:
[ y dy e^{-x} dx ]
Separating the variables:
[ y dy e^{-x} dx ]
Integrating both sides:
[ int y dy int e^{-x} dx ]
Solving the integrals:
[ frac{y^2}{2} -e^{-x} C ]
Simplifying:
[ y^2 -2e^{-x} 2C ]
This is another form of the solution, where ( C ) is a constant of integration.
Conclusion
The differential equation ( ln y frac{dy}{dx} -x ) can be solved in multiple ways, leading to similar results. The methods involve separation of variables and integration by parts, resulting in a general solution that can be written in different but equivalent forms.