(3xy^2-y^3)dx-(2x^2y-xy^2)dy=0

Jasonbradley

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Jun 16, 2024

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Solving the Differential Equation (3xy^2 - y^3)dx - (2x^2y - xy^2)dy = 0

This article will walk through the process of solving the given differential equation:

(3xy^2 - y^3)dx - (2x^2y - xy^2)dy = 0

Identifying the Type of Differential Equation

First, we need to identify the type of differential equation we're dealing with. This equation is a first-order homogeneous differential equation. Here's why:

• First-order: The highest derivative present is the first derivative (dy/dx).
• Homogeneous: All terms in the equation have the same degree (3 in this case).

Solving the Homogeneous Equation

To solve homogeneous equations, we use the substitution:

y = vx

where 'v' is a function of x. This substitution helps simplify the equation and makes it easier to solve.

1. Substitute y = vx and dy = vdx + xdv

Substituting these into the original equation gives us:

(3x(vx)^2 - (vx)^3)dx - (2x^2(vx) - x(vx)^2)(vdx + xdv) = 0

2. Simplify the Equation

Simplifying the equation, we get:

(3v^2x^3 - v^3x^3)dx - (2v^2x^3 - v^2x^3)(vdx + xdv) = 0

Combining like terms:

(3v^2x^3 - v^3x^3 - 2v^3x^3 + v^3x^3)dx - (2v^3x^4 - v^3x^4)dv = 0

Further simplification:

(3v^2x^3 - 2v^3x^3)dx - (v^3x^4)dv = 0

3. Separate the Variables

Now, separate the variables x and v:

(3v^2x^3 - 2v^3x^3) / (v^3x^4) dx = dv

Simplifying:

(3/v - 2/x) dx = dv

4. Integrate Both Sides

Integrate both sides with respect to their respective variables:

∫(3/v - 2/x) dx = ∫dv

This gives us:

3ln|v| - 2ln|x| = v + C

5. Substitute Back y = vx

Substitute back v = y/x:

3ln|y/x| - 2ln|x| = y/x + C

6. Simplify the Solution

The solution can be further simplified using logarithmic properties:

ln|y^3/x^5| = y/x + C

This is the general solution to the differential equation (3xy^2 - y^3)dx - (2x^2y - xy^2)dy = 0.

Finding a Particular Solution

To find a particular solution, you would need an initial condition (e.g., y(1) = 2). This initial condition allows you to solve for the constant C in the general solution.

This step-by-step process demonstrates how to solve a first-order homogeneous differential equation. Remember that the specific solution will depend on the initial conditions provided.