Solving the Differential Equation
Introduction
Understanding and solving differential equations is a fundamental skill in mathematics, especially when dealing with real-world applications in physics, engineering, and economics. This article aims to provide a comprehensive solution for the differential equation (frac{dy}{dx} sqrt{y} - sqrt{y} sin x). We will explore the method of separation of variables and integration to find the general solution.
Problem Statement
The given differential equation is:
[ frac{dy}{dx} sqrt{y} - sqrt{y} sin x ]Factoring the Right-Hand Side
We can factor out (sqrt{y}) from the right-hand side of the equation:
[ frac{dy}{dx} sqrt{y}(1 - sin x) ]This suggests that we can use the method of separation of variables.
Separation of Variables
We rewrite the equation by separating the variables (y) and (x):
[ frac{dy}{sqrt{y}} (1 - sin x) dx ]The equation now suggests that the derivative of (y) with respect to (x) can be integrated on both sides.
Integration of Both Sides
First, we integrate the left-hand side:
[ int frac{dy}{sqrt{y}} int 2sqrt{y} dy 2sqrt{y} C_1 ]Next, we integrate the right-hand side:
[ int (1 - sin x) dx x - cos x C_2 ]Combining the two results, we get:
[ 2sqrt{y} x - cos x C ]Where (C C_2 - C_1) is the constant of integration.
Solving for (y)
To solve for (y), we isolate (sqrt{y}) and then square both sides:
[ sqrt{y} frac{x - cos x C}{2} ] [ y left(frac{x - cos x C}{2}right)^2 ]Thus, the general solution is:
[ y frac{(x - cos x C)^2}{4} ]Alternative Approach
An alternative method is to observe that (dy/sqrt{y} dx(1 - sin x)). Integrating both sides gives:
[ 2sqrt{y} x - cos x C ]Solving for (sqrt{y}) and then squaring both sides, we get the same general solution:
[ sqrt{y} frac{x - cos x C}{2} ] [ y left(frac{x - cos x C}{2}right)^2 ]Conclusion
In conclusion, the general solution to the differential equation (frac{dy}{dx} sqrt{y} - sqrt{y} sin x) is given by:
[ y frac{(x - cos x C)^2}{4} ]This solution is valid for (y > 0), and (C) is an arbitrary constant.
Further Reading
For deeper understanding of differential equations and techniques for solving them, you can explore the following resources:
Differential Equations: An Introduction to Modern Methods and Applications Introduction to Ordinary Differential Equations Separation of Variables and Applications