Solving the Differential Equation dy/dx cos(y) - x 1
Understanding and solving differential equations is a crucial part of mathematical analysis, particularly in fields such as physics, engineering, and economics. One such differential equation is dy/dx cos(y) - x 1, which can be tackled through a series of steps involving manipulation and integration. This article will guide you through the process of solving this equation step by step, providing a detailed explanation and highlighting key concepts.
Step-by-Step Solution
To solve the differential equation dy/dx cos(y) - x 1, we begin by rewriting it in a form suitable for integration. This equation can be simplified and solved using the method of separation of variables. Let us walk through this method in detail.
1. Separation of Variables
The goal is to separate the variables y and x such that we can integrate each side separately.
Starting with the given equation:
$$frac {dy}{cosy-x 1}dfrac {dx}1dfrac {dy-dx}{cosy-x 1-1}dfrac {dy-x}{cosy-x }$$By taking the second and fourth ratios and substituting, we get:
$$dxsec y-x dy-x$$Integrating both sides of the equation is the next step. The integral of the left side is:
$$ int dx x C_1 $$where C_1 is the constant of integration. The right side requires a bit more handling.
2. Integration
Let's integrate the right side of the equation:
$$ int sec y - x , dy - x , dx x C_1 $$
After integration, we get:
$$ x ln|sec y - x| - x tan y - x C $$where C C_1 is the constant of integration. Simplifying the equation, we obtain:
$$ x ln|sec y - x| - x tan y C $$Conclusion
Therefore, the solution to the differential equation dy/dx cos(y) - x 1 is:
$$ x ln|sec y - x| - x tan y C $$This step-by-step approach highlights the importance of careful manipulation and integration in solving differential equations. Understanding these techniques not only helps in solving specific problems but also aids in grasping the broader concepts of mathematical analysis.
Keywords
Solution, Integration, Differential Equation