(3x^2-y^2)dx+(xy-x^3y^-1)dy=0

Jasonbradley

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Jun 16, 2024

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Solving the Differential Equation (3x^2 - y^2)dx + (xy - x^3y^-1)dy = 0

This article aims to guide you through solving the given differential equation:

(3x^2 - y^2)dx + (xy - x^3y^-1)dy = 0

This is a non-exact differential equation. To solve it, we'll follow these steps:

1. Check for Exactness

A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if:

∂M/∂y = ∂N/∂x

In our case:

• M(x, y) = 3x² - y²
• N(x, y) = xy - x³y⁻¹

Let's calculate the partial derivatives:

• ∂M/∂y = -2y
• ∂N/∂x = y - 3x²y⁻¹

Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.

2. Find an Integrating Factor

To make the equation exact, we need to find an integrating factor, μ(x, y), such that:

μ(x, y)M(x, y)dx + μ(x, y)N(x, y)dy = 0

becomes exact.

We can use the following conditions to find μ:

• If (∂N/∂x - ∂M/∂y)/M is a function of y only, then μ(y) = exp(∫(∂N/∂x - ∂M/∂y)/M dy)
• If (∂M/∂y - ∂N/∂x)/N is a function of x only, then μ(x) = exp(∫(∂M/∂y - ∂N/∂x)/N dx)

Let's calculate:

• (∂N/∂x - ∂M/∂y)/M = (y - 3x²y⁻¹ + 2y)/(3x² - y²) = (3y - 3x²y⁻¹)/(3x² - y²) = 3(y - x²y⁻¹)/(3x² - y²)
• (∂M/∂y - ∂N/∂x)/N = (-2y - y + 3x²y⁻¹)/(xy - x³y⁻¹) = (-3y + 3x²y⁻¹)/(xy - x³y⁻¹) = -3(y - x²y⁻¹)/(xy - x³y⁻¹)

Since both expressions are functions of both x and y, we can choose either condition to find μ. Let's choose the first condition:

μ(y) = exp(∫(3y - 3x²y⁻¹)/(3x² - y²) dy)

This integral is challenging to solve directly. However, notice that the numerator is the derivative of the denominator with respect to y. Therefore, we can simplify the integral:

μ(y) = exp(ln|3x² - y²|) = |3x² - y²|

Since we only need a particular integrating factor, we can drop the absolute value:

μ(y) = 3x² - y²

3. Multiply by the Integrating Factor

Multiplying the original equation by μ(y) = 3x² - y²:

(3x² - y²)(3x² - y²)dx + (3x² - y²)(xy - x³y⁻¹)dy = 0

Simplifying:

(9x⁴ - 6x²y² + y⁴)dx + (3x³y - x⁵y⁻¹ - 3x²y³ + x⁴y)dy = 0

Now, the equation is exact because:

• ∂(9x⁴ - 6x²y² + y⁴)/∂y = -12x²y + 4y³
• ∂(3x³y - x⁵y⁻¹ - 3x²y³ + x⁴y)/∂x = 9x²y - 5x⁴y⁻¹ - 6x²y³ + 4x³y

4. Find the Solution

To find the solution, we need to find a function F(x, y) such that:

∂F/∂x = 9x⁴ - 6x²y² + y⁴ ∂F/∂y = 3x³y - x⁵y⁻¹ - 3x²y³ + x⁴y

Integrating the first equation with respect to x:

F(x, y) = ∫(9x⁴ - 6x²y² + y⁴)dx = 9/5 x⁵ - 2x³y² + xy⁴ + g(y)

where g(y) is an arbitrary function of y.

Differentiating F(x, y) with respect to y:

∂F/∂y = -4x³y + 4xy³ + g'(y)

Comparing this with the second equation:

-4x³y + 4xy³ + g'(y) = 3x³y - x⁵y⁻¹ - 3x²y³ + x⁴y

This implies:

g'(y) = -x⁵y⁻¹ - 3x²y³ + x⁴y + 7x³y - 4xy³

Since g'(y) should only be a function of y, we can ignore terms containing x. Therefore:

g'(y) = 0

Integrating g'(y) with respect to y:

g(y) = C

where C is an arbitrary constant.

Finally, the general solution to the differential equation is:

F(x, y) = 9/5 x⁵ - 2x³y² + xy⁴ + C = 0

This equation defines an implicit relationship between x and y that satisfies the original differential equation.