Solving the Differential Equation (y^2 frac{dy}{dx} 5y) Through Variable Separation
In mathematics, the differential equation can be a challenging topic for many students and researchers due to its abstract nature and the variety of techniques required to solve various types. One common technique is called variable separation, which simplifies the equation into more manageable parts. Let's explore how to solve the differential equation (y^2 frac{dy}{dx} 5y) using this method.
Step-by-Step Solution
1. Start with the given equation:
[y^2 frac{dy}{dx} 5y]
2. Separate the variables: To separate the variables, we divide both sides by (y^2) and multiply both sides by (dx).
[frac{y^2}{y} dy 5 dx]
3. Integrate both sides: We integrate the left-hand side with respect to (y) and the right-hand side with respect to (x).
[int frac{y^2}{y} dy int 5 dx]
4. Simplify the integrals: On the left-hand side, we can simplify the expression.
[int y dy int 5 dx]
5. Perform the integration: We know that the integral of (y dy) is (frac{y^2}{2}), and the integral of (5 dx) is (5x C_0), where (C_0) is the constant of integration.
[frac{y^2}{2} 5x C_0]
6. Write the final solution: We can finally express the solution in a boxed format for clarity.
[x frac{1}{10}y^2 - frac{2}{5} ln y C]
Explanation of the Variables Separation Method
The variable separation method involves splitting a differential equation into two separate integrals, one with respect to (y) and the other with respect to (x). This enables us to integrate each part independently and find a relationship between (x) and (y).
It's important to note that after integrating, the solution might appear in a form involving logarithms or other functions. The constant of integration is often denoted as (C) and can be adjusted according to the initial conditions of the problem.
Further Exploration
For more complex differential equations, variable separation might not always be the most straightforward method. Other techniques such as integrating factors, substitution methods, or numerical methods might be more appropriate. However, for simpler equations like (y^2 frac{dy}{dx} 5y), variable separation is an efficient and effective approach.
Understanding and mastering these techniques will greatly enhance your ability to solve a wide range of differential equations, which are fundamental in various fields such as physics, engineering, and applied mathematics.