Solving Homogeneous Differential Equations: A Step-by-Step Guide Using the Example $x^3 - y^3 dx xy^2 dy$
Homogeneous differential equations are a class of differential equations that can be solved by specific techniques. In this article, we explore a detailed step-by-step guide on solving the equation x3 - yx3 dx xy2 dy. The steps include substitution, separation of variables, and integration to find the general solution. Let's dive into the process in detail.
1. Initial Equation and Substitution
The given differential equation can be written as:
[x^3 - yx^3 dy xy^2 dx]Let yxv and mathbf{dy} x dv v dx. Substituting mathbf{dy} v dx x dv into the original equation:
[x^3 - x^3 v^3 (v dx x dv) x^3 v^2 dx]Simplifying:
[x^3 - x^3 v^3 v dx - x^4 v^3 dv x^3 v^2 dx] [x^3 - x^3 v^4 dx - x^4 v^3 dv x^3 v^2 dx]Rearranging and simplifying further:
[begin{align}-x^3 v^4 - 1 x^3 x dv x^4 v^2 dvend{align}]Dividing both sides by mathbf{-2v^3 - 1/v^2}:
[begin{align}text{Right side:} frac{x^3 x dv x^4 v^2 dv}{-2 v^3 - 1/x v^2} frac{1 - 2v^3}{v^2} end{align}]The equation now becomes:
[begin{align}frac{dv}{dx} frac{1 - 2v^3}{v^2}end{align}]Separating variables:
[begin{align}int frac{v^2 dv}{1 - 2v^3} int frac{dx}{x}end{align}]2. Integration and Solving for $v$
Let's integrate left side first:
[ int frac{v^2 dv}{1 - 2v^3} -frac{1}{6} ln|1 - 2v^3|]The right side integration is straightforward:
[ int frac{dx}{x} ln|x| C_1]Combining both sides:
[begin{align}-frac{1}{6} ln|1 - 2v^3| ln|x| C_1end{align}]Multiplying by 6 and exponentiating both sides:
[begin{align}ln|1 - 2v^3|^{-1/6} ln|x| C_11 - 2v^3 -x^6 C^6end{align}]Since mathbf{v} frac{y}{x}, we have:
[ 1 - 2left(frac{y}{x}right)^3 -frac{x^6}{C^6} ]Combining and solving for mathbf{y}:
[begin{align}y pm frac{x^3}{sqrt[3]{C_1 / x^6} / 2}end{align}]Consolidating the constants for a final form:
[ y frac{x^6}{C_1 / 2}]This represents the general solution to the given differential equation, accounting for the arbitrary constant C.
Conclusion
This method of solving homogeneous differential equations involves several steps, including substitution, separation of variables, and integration. By following these steps, one can solve a wide range of homogeneous differential equations. Understanding the underlying principles and techniques is crucial for tackling more complex equations in advanced calculus and differential equations coursework.