(x2 + 2y2) Dx Dy = Xy Y(−1) = 3

4 min read Jun 17, 2024

Solving the Differential Equation: (x² + 2y²) dx dy = xy, y(-1) = 3

This problem involves a differential equation with an initial condition, which means we need to find a function that satisfies both the equation and the given point. Let's break down the solution step-by-step:

1. Identifying the Type of Equation

The given equation is a non-linear first-order differential equation. We can see this because:

Non-linear: It involves terms with powers of x and y higher than 1.
First-order: The highest derivative present is the first derivative (dy/dx).

2. Solving the Differential Equation

Unfortunately, this particular equation doesn't have a straightforward solution using standard methods like separation of variables or integrating factors. Here's why:

No clear separation: We cannot easily separate the x and y terms on opposite sides of the equation.
Non-exact: The equation doesn't satisfy the condition for being exact, which would allow for a specific integration approach.

Therefore, we might need to consider numerical methods or approximation techniques to solve this differential equation.

3. Understanding the Initial Condition

The initial condition y(-1) = 3 tells us that the solution curve passes through the point (-1, 3). This condition is crucial for finding a specific solution instead of a general solution.

4. Numerical Methods (Example)

One way to approach this is using a numerical method like the Euler method. Here's a brief outline:

Discretize the domain: Divide the x-axis into small intervals.
Approximate the derivative: Use the slope at the current point to estimate the change in y over the interval.
Iterate: Repeat the process using the new point and its slope, moving along the curve.

The accuracy of the solution depends on the step size used in the discretization. Smaller step sizes lead to greater accuracy but require more calculations.

5. Finding a Solution

Due to the complexity of the equation, we cannot provide an exact closed-form solution in this response. However, we can outline the key steps and highlight the challenges involved.

To find a solution, you would need to:

Choose a numerical method: Euler, Runge-Kutta, etc.
Implement the method: Use a programming language or software tool.
Set the initial condition: y(-1) = 3
Run the algorithm: Obtain a numerical approximation of the solution curve.

Note: The specific approach and choice of method might depend on the desired level of accuracy and computational resources available.

In conclusion, while this differential equation presents a challenge due to its non-linear nature, numerical methods provide a powerful tool for finding approximate solutions and gaining insights into the behavior of the solution curve.