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Solving the Differential Equation (x^2 + 2xy - y^2)dx + (y^2 + 2xy - x^2)dy = 0
This article will explore the solution to the given differential equation:
(x^2 + 2xy - y^2)dx + (y^2 + 2xy - x^2)dy = 0
We'll employ a systematic approach to solve this equation, which is a first-order homogeneous differential equation.
Identifying the Type of Equation
The given equation is a homogeneous differential equation because it can be rewritten in the form:
M(x, y)dx + N(x, y)dy = 0
where both M(x, y) and N(x, y) are homogeneous functions of the same degree. In this case, both M(x, y) and N(x, y) are homogeneous functions of degree 2.
Solving the Homogeneous Differential Equation
To solve this type of equation, we use the substitution y = vx, where v is a function of x. This substitution helps us transform the equation into a separable differential equation.
1. Substitution:
Substituting y = vx, we get:
- dy = vdx + xdv
2. Substituting into the Original Equation:
Substituting the values of y and dy into the original equation, we get:
(x^2 + 2x(vx) - (vx)^2)dx + ((vx)^2 + 2x(vx) - x^2)(vdx + xdv) = 0
3. Simplifying the Equation:
Simplifying the equation, we get:
(x^2 + 2vx^2 - v^2x^2)dx + (v^2x^2 + 2vx^2 - x^2)(vdx + xdv) = 0
4. Separating the Variables:
Rearranging the terms and factoring out common factors, we get:
(1 + 2v - v^2)dx + (v^2 + 2v - 1)xdv = 0
Dividing both sides by x(1 + 2v - v^2)(v^2 + 2v - 1), we get:
dx/x + (v^2 + 2v - 1)/(1 + 2v - v^2)dv = 0
5. Integrating Both Sides:
Integrating both sides of the equation, we get:
ln|x| - ln|(1 + 2v - v^2)| = C
where C is the constant of integration.
6. Simplifying and Solving for v:
Simplifying the equation and solving for v, we get:
v = (1 ± √(5))/2
7. Back-Substitution:
Substituting back y = vx, we get:
y = ((1 ± √(5))/2)x
General Solution
Therefore, the general solution to the differential equation (x^2 + 2xy - y^2)dx + (y^2 + 2xy - x^2)dy = 0 is:
y = ((1 ± √(5))/2)x
This solution represents a family of straight lines passing through the origin, with slopes of (1 + √(5))/2 and (1 - √(5))/2.
Conclusion
We successfully solved the given differential equation using the method of substitution and integration. The solution highlights the importance of identifying the type of differential equation and applying the appropriate techniques for solving it. This example provides a comprehensive understanding of solving first-order homogeneous differential equations.