(x^3-3xy^2)dx=(y^3-3x^2y)dy

4 min read Jun 17, 2024

Solving the Differential Equation: (x^3 - 3xy^2)dx = (y^3 - 3x^2y)dy

This article will guide you through the process of solving the given differential equation. We'll explore the techniques involved and arrive at a general solution.

Identifying the Type of Differential Equation

The equation (x^3 - 3xy^2)dx = (y^3 - 3x^2y)dy is a homogeneous differential equation. This is because we can rewrite it in the form:

dy/dx = f(y/x)

To see this, divide both sides by (x^3 - 3xy^2) and by dx:

dy/dx = (y^3 - 3x^2y) / (x^3 - 3xy^2)

Now, divide both the numerator and denominator by x^3:

dy/dx = ((y/x)^3 - 3(y/x)) / (1 - 3(y/x)^2)

This clearly shows that the right-hand side is a function of (y/x).

Solving Homogeneous Equations

We solve homogeneous equations using the substitution u = y/x. This leads to:

y = ux
dy/dx = u + x(du/dx)

Substituting these into the original equation:

u + x(du/dx) = (u^3 - 3u) / (1 - 3u^2)

Rearranging the equation to separate variables:

x(du/dx) = (u^3 - 3u) / (1 - 3u^2) - u

x(du/dx) = (u^3 - 3u - u + 3u^3) / (1 - 3u^2)

x(du/dx) = (4u^3 - 4u) / (1 - 3u^2)

Simplifying:

(1 - 3u^2) du / (4u^3 - 4u) = dx/x

Now we have the variables separated, ready for integration.

Integration and Solution

Integrating both sides:

∫(1 - 3u^2) du / (4u^3 - 4u) = ∫dx/x

The left-hand side integral requires partial fraction decomposition. After solving for the coefficients and integrating, we get:

(1/4)ln|u| + (1/8)ln|1 - 3u^2| = ln|x| + C

Substituting back u = y/x:

(1/4)ln|y/x| + (1/8)ln|1 - 3(y/x)^2| = ln|x| + C

This is the general solution to the differential equation. The solution can be further manipulated for a more compact or readable form.

Conclusion

By identifying the differential equation as homogeneous and using the appropriate substitution, we were able to solve the equation. The general solution involves a combination of logarithmic terms. The solution can be simplified and potentially used to analyze the behavior of the system described by the differential equation.