(1-x^2)(1-y)dx=xy(1+y)dy

4 min read Jun 16, 2024

Solving the Differential Equation: (1-x^2)(1-y)dx = xy(1+y)dy

This article will guide you through solving the given differential equation: (1-x^2)(1-y)dx = xy(1+y)dy. This equation is classified as a separable differential equation because we can rearrange it so that all terms containing x and dx are on one side, and all terms containing y and dy are on the other.

Step 1: Separating the Variables

Divide both sides by (1-y) and xy(1+y):
```
(1-x^2)/(xy(1+y)) dx = dy/(1-y)
```

Simplify:

(1/(xy(1+y)) - x/(y(1+y))) dx = dy/(1-y)

Now the equation is separated.

Step 2: Integrating Both Sides

Integrate the left side with respect to x:
```
∫ (1/(xy(1+y)) - x/(y(1+y))) dx = ∫ dy/(1-y)
```
To solve the integrals on the left side, we'll use partial fractions:
- 1/(xy(1+y)): We can write this as A/x + B/y + C/(1+y) and solve for A, B, and C.
- x/(y(1+y)): We can write this as D/y + E/(1+y) and solve for D and E.
After solving for the constants and integrating, the left side becomes:
```
(1/y)ln|x| - (1/(1+y))ln|1+x| + C1 
```
Integrate the right side with respect to y:
```
∫ dy/(1-y) = -ln|1-y| + C2
```

Step 3: Combining Constants and Solving for y

Combine the constants of integration: C1 - C2 = C

The general solution is:

(1/y)ln|x| - (1/(1+y))ln|1+x| = -ln|1-y| + C

This equation implicitly defines y in terms of x. It's difficult to explicitly solve for y in this case.

Conclusion

We have successfully solved the differential equation (1-x^2)(1-y)dx = xy(1+y)dy and obtained an implicit solution. This solution can be used to analyze the behavior of the dependent variable y with respect to the independent variable x.

Remember that the solution includes an arbitrary constant C. This constant represents different possible solutions to the differential equation, creating a family of curves.