Solving Ordinary Differential Equations Through Symmetry and Homogeneity

In the realm of differential equations, understanding the underlying symmetries can significantly aid in finding solutions. This article explores two such techniques: leveraging Lie symmetries for solving ordinary differential equations (ODEs) and employing the method of homogeneity. We will delve into these methods with specific examples to illustrate their application.

Leveraging Lie Symmetries for Solving ODEs

Lie symmetries are a powerful tool in the study of differential equations. They are transformations that leave the differential equation invariant. For an ODE, if a transformation can be applied to the variables such that the resulting equation remains unchanged, it indicates a Lie symmetry. This symmetry can transform the ODE into a simpler form, often leading to an integrable equation.

The Problem and Solution

Consider the problem:

$$frac{dy}{dx} x^2y^2$$

This is a homogeneous ODE. A function y(x) is called homogeneous if it satisfies the property y(λx) λny(x)

To solve this, we use the method of substitution. Let us put y(x) xv(x). Then, dy/dx xv'( x) v(x). Substituting this into the ODE, we get:

$$xv' v x^3v^2$$

Dividing both sides by xv, we have:

$$frac{1}{v} frac{1}{x} frac{x^2v^2}{v}$$

Let v p. Then, we get:

$$frac{1}{p} frac{1}{x} x^2p^2$$

Subtracting 1/p from both sides:

$$frac{1}{p^2} x^2p - frac{1}{x}$$

Thus:

$$frac{1}{p^2} x^2p - frac{1}{x}$$

Take the differential:

$$frac{dp}{dx} -frac{2}{x^3} frac{1}{x^2}p^2$$

This is a linear in p differential equation and can be solved by standard methods.

Understanding and Applying Lie Symmetries

Now, consider another example:

$$frac{dy}{dx} x^2y^2$$

By recognizing the equation as a homogeneous equation, we can use the substitution y px. Then, dy/dx xp' p. Substituting into the ODE:

$$xp' p x^3p^2$$

Dividing both sides by xp (assuming x ≠ 0):

$$frac{1}{p} frac{1}{x} x^2p$$

Let v p. Then, we have:

$$frac{1}{v} frac{1}{x} x^2v^2$$

Rewriting this as a separable differential equation:

$$frac{dp}{dx} x^2p - frac{1}{x}$$

This can be solved using standard techniques for separable differential equations. The solution can be found by integrating both sides.

Extending to Lie Symmetry Solution

For the ODE where y(x) λx, and y(λx) λy(x), we apply Lie symmetries to find a more general solution. By introducing the invariant u y/x, we transform the equation into:

$$frac{du}{1-u^2} frac{dx}{2}$$

This is a much simpler form to solve. Integrating both sides:

$$int frac{du}{1-u^2} int frac{dx}{2}$$

The left side is a standard integral that can be solved using partial fractions:

$$frac{1}{2} int left( frac{1}{1 u} frac{1}{1-u} right) du frac{1}{2} ln left| frac{1 u}{1-u} right| frac{x}{2} C$$

Thus:

$$frac{1 u}{1-u} Ke^x$$

Substituting back u y/x, we get:

$$frac{1 frac{y}{x}}{1-frac{y}{x}} Ke^x$$

Simplifying, we obtain:

$$frac{x y}{x-y} Ke^x$$

Rewriting, we get the solution:

$$y x cdot frac{CJ_{3/4}(frac{x^2}{2}) - J_{-3/4}(frac{x^2}{2})}{CJ_{-1/4}(frac{x^2}{2}) J_{1/4}(frac{x^2}{2})}$$

This solution can be further simplified by recognizing specific recurrence relations and applying transformations to avoid singularities.

Conclusion

Solving differential equations through symmetries and homogeneity is a valuable technique that simplifies complex problems. By leveraging Lie symmetries and employing the method of homogeneity, one can tackle a wide range of ODEs and find explicit solutions. Understanding these methods enhances the ability to analyze and solve differential equations in various applications.