(1-x^2)dy/dx=2y

4 min read Jun 16, 2024

Solving the Differential Equation (1-x^2)dy/dx = 2y

This article explores the solution to the differential equation (1-x^2)dy/dx = 2y. We will delve into the steps of solving this equation, understanding its type, and analyzing its solution.

Understanding the Differential Equation

The given equation is a first-order linear ordinary differential equation. It is classified as:

First-order: Because the highest derivative appearing in the equation is the first derivative (dy/dx).
Linear: Because the dependent variable (y) and its derivative appear only in a linear form, meaning they are not raised to any power or multiplied together.
Ordinary: Because the equation involves only ordinary derivatives, not partial derivatives.

Solving the Differential Equation

To solve the equation, we will follow these steps:

Rearrange the equation:

Divide both sides of the equation by (1-x^2)y:
```
dy/y = 2dx / (1-x^2)
```
Integrate both sides:

Integrate both sides of the equation:
```
∫ dy/y = ∫ 2dx / (1-x^2) 
```
The left side integrates to ln|y| and the right side can be integrated using partial fractions. The integral becomes:
```
ln|y| = ln|1+x| - ln|1-x| + C
```
Solve for y:

Combine the logarithms on the right side and exponentiate both sides:
```
y = e^(ln|1+x| - ln|1-x| + C)
```
Simplifying, we get:
```
y = C(1+x)/(1-x)
```
Where C = e^C is a constant of integration.

Analyzing the Solution

The solution to the differential equation is y = C(1+x)/(1-x), where C is an arbitrary constant. This solution represents a family of curves that are solutions to the differential equation. The value of C determines the specific curve within this family.

The solution exhibits the following characteristics:

Singular points: The solution has a singularity at x = 1, where the denominator becomes zero. This indicates that the solution is not defined at x = 1.
Asymptotes: As x approaches ±1, the solution approaches infinity, implying vertical asymptotes at x = ±1.
Symmetry: The solution is not symmetric about the y-axis or the origin, indicating a lack of even or odd symmetry.

Conclusion

By following the steps outlined above, we have solved the differential equation (1-x^2)dy/dx = 2y. We have found the general solution y = C(1+x)/(1-x), which represents a family of curves that satisfy the equation. The solution exhibits specific characteristics like singularities and asymptotes. Understanding these characteristics provides a deeper insight into the behavior of the solution.