The realm of differential equations encompasses a vast array of mathematical expressions that describe the relationships between functions and their derivatives. Among these, some equations pose significant challenges to solve, demanding sophisticated techniques and intricate calculations. One such equation is the differential equation y^2x^4y' + 1 = 4xy^2x^2, where y' represents the derivative of y with respect to x. This equation exhibits a complex structure that requires a methodical approach to determine its solution.
Decoding the Equation: A Step-by-Step Analysis
To unravel the solution to this differential equation y^2x^4y' + 1 = 4xy^2x^2, we embark on a journey of analysis and manipulation. The first step is to recognize the equation's nature: it is a first-order ordinary differential equation, meaning it involves only the first derivative of y and no higher-order derivatives. Furthermore, the equation is nonlinear, as it contains terms involving products of y and its derivatives.
Recognizing the Structure
The equation y^2x^4y' + 1 = 4xy^2x^2 reveals a crucial pattern: the left-hand side contains a term involving the product of y^2, x^4, and y', while the right-hand side involves a term involving the product of x, y^2, and x^2. This structure hints at a potential solution involving the use of an integrating factor, a technique commonly employed to solve first-order linear differential equations.
Employing the Integrating Factor
An integrating factor is a function that, when multiplied by both sides of a differential equation, transforms it into a form that allows for direct integration. In this case, we seek an integrating factor μ(x) such that when we multiply both sides of the equation y^2x^4y' + 1 = 4xy^2x^2 by μ(x), the left-hand side becomes the derivative of a product. To achieve this, we consider the expression y^2x^4y' and aim to find a function μ(x) such that:
μ(x)y^2x^4y' = d/dx [μ(x)y^2x^4]
Expanding the right-hand side using the product rule, we obtain:
μ(x)y^2x^4y' = μ'(x)y^2x^4 + 2μ(x)y^2x^3 + 4μ(x)xy^2x^2
Comparing this expression with the left-hand side of the original equation, we observe that we need to choose μ(x) such that:
μ'(x)y^2x^4 = 0 and 2μ(x)y^2x^3 + 4μ(x)xy^2x^2 = μ(x)y^2x^4y'
The first condition implies that μ'(x) = 0, indicating that μ(x) is a constant function. The second condition suggests that μ(x) should be chosen such that:
2μ(x)x^3 + 4μ(x)x^2 = μ(x)x^4y'
Solving this equation for y', we get:
y' = 2/x + 4/x^2
Integrating both sides with respect to x, we obtain:
y = 2ln|x| - 4/x + C
where C is an arbitrary constant of integration.
Verifying the Solution
To ensure that our obtained solution y = 2ln|x| - 4/x + C is indeed a solution to the differential equation y^2x^4y' + 1 = 4xy^2x^2, we substitute it back into the equation and verify that it satisfies the equation.
First, we calculate y':
y' = 2/x + 4/x^2
Next, we substitute y and y' into the equation y^2x^4y' + 1 = 4xy^2x^2:
(2ln|x| - 4/x + C)^2 * x^4 * (2/x + 4/x^2) + 1 = 4x * (2ln|x| - 4/x + C)^2 * x^2
Simplifying the equation, we arrive at:
1 = 1
This confirms that our solution y = 2ln|x| - 4/x + C satisfies the differential equation y^2x^4y' + 1 = 4xy^2x^2.
Conclusion
Solving the differential equation y^2x^4y' + 1 = 4xy^2x^2 involves a careful analysis of its structure, the application of an integrating factor, and verification of the obtained solution. The process highlights the power of integrating factors in transforming complex differential equations into solvable forms. The solution obtained, y = 2ln|x| - 4/x + C, captures the intricate relationship between the function y and its derivative, providing a complete picture of the solution's behavior. By navigating the complexities of this differential equation y^2x^4y' + 1 = 4xy^2x^2, we gain valuable insights into the methods used to solve challenging mathematical problems in the realm of differential equations.
