(3x+4y-5)^(2)(dy)/(dx)=a^(2)

Jasonbradley

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Jun 16, 2024

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Solving the Differential Equation (3x + 4y - 5)^2 (dy/dx) = a^2

This article will guide you through the steps of solving the differential equation:

(3x + 4y - 5)^2 (dy/dx) = a^2

1. Understanding the Equation

The equation is a first-order, non-linear differential equation. It's considered non-linear due to the presence of the squared term (3x + 4y - 5)^2.

2. Simplifying the Equation

We can simplify the equation by dividing both sides by (3x + 4y - 5)^2:

(dy/dx) = a^2 / (3x + 4y - 5)^2

3. Recognizing the Form

The equation now resembles a separable differential equation. Separable differential equations can be solved by separating the variables (x and y) and integrating both sides.

4. Separating the Variables

To separate the variables, multiply both sides by dx and divide both sides by (3x + 4y - 5)^2:

(dy)/(3x + 4y - 5)^2 = (a^2/dx)

5. Integration

Integrate both sides of the equation:

• Left Side: This side requires a substitution. Let u = (3x + 4y - 5). Then du = (3 + 4(dy/dx)) dx. Rearranging, we get (dy/dx) = (du/4) - (3/4). Substituting this back into the integral, we get:

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

  Integrating both terms:

  -1/(4u) - (3/4)x + C1 = 0

• Right Side: Integrating the right side is straightforward:

  ∫ (a^2/dx) = a^2x + C2

6. Combining the Results

Combining the results from both sides, we get:

-1/(4(3x + 4y - 5)) - (3/4)x + C1 = a^2x + C2

7. Solving for y

To solve for y, we need to simplify the equation and isolate y:

• Combine constants: Let C = C2 - C1.
• Multiply both sides by -4(3x + 4y - 5): 1 + 3x(3x + 4y - 5) - 4C(3x + 4y - 5) = -4a^2x(3x + 4y - 5)
• Expand and simplify: 1 + 9x^2 + 12xy - 15x - 12Cx - 16Cy + 20C = -12a^2x^2 - 16a^2xy + 20a^2x
• Group y terms: (12x + 16C - 16a^2x)y = -9x^2 + 15x - 20C - 1 - 12a^2x^2 - 20a^2x
• Solve for y: y = (-9x^2 + 15x - 20C - 1 - 12a^2x^2 - 20a^2x) / (12x + 16C - 16a^2x)

8. General Solution

The general solution to the differential equation is:

y = (-9x^2 + 15x - 20C - 1 - 12a^2x^2 - 20a^2x) / (12x + 16C - 16a^2x)

where C is an arbitrary constant.

Important Note: The general solution might be further simplified or rearranged depending on the specific context or desired form of the solution.