Solving a First Order Differential Equation with Degree 2
When dealing with differential equations, particularly first-order equations with degree 2, a systematic approach can help in finding their general solutions. In this article, we will explore the process of solving a specific first-order differential equation with degree 2, providing a detailed step-by-step guide.
Introduction to the Problem
The differential equation we will be working with is:
xy^2 left( frac{dy}{dx} right)^2 - 2y^3 cdot frac{dy}{dx} - x^3 2xy^2 0
Our goal is to find the general solution to this equation without plugging in specific values for any variables.
Step-by-Step Solution
Divide by (xy^2): Begin by dividing both sides of the equation by (xy^2).
[ left( frac{dy}{dx} right)^2 - frac{2y}{x} cdot frac{dy}{dx} - left( frac{x^2}{y^2} - 2 right) 0 ]
Apply the Quadratic Formula: We can rewrite the equation in the standard form of a quadratic equation.
[ left( frac{dy}{dx} right)^2 - frac{2y}{x} cdot frac{dy}{dx} - left( frac{x^2}{y^2} - 2 right) 0 ]
The quadratic formula gives us:
[ frac{dy}{dx} frac{frac{2y}{x} pm sqrt{left( frac{2y}{x} right)^2 - 4 left( frac{x^2}{y^2} - 2 right)}}{2} ]
Simplify the Expression: Further simplification yields:
[ frac{dy}{dx} frac{y}{x} pm frac{sqrt{y^4 - x^4 - 2x^2y^2}}{xy} ]
Substitution Method: Consider the substitution (y vx). Then, (frac{dy}{dx} v x frac{dv}{dx}).
Substitute and simplify:
[ v x frac{dv}{dx} v pm frac{sqrt{y^4 - x^4 - 2x^2y^2}}{xv} ]
This simplifies further:
[ x frac{dv}{dx} pm frac{sqrt{y^4 - x^4 - 2x^2y^2}}{xv} ]
Separation of Variables: Separate the variables and integrate both sides.
[ frac{v}{v^2 - 1} dv pm frac{dx}{x} ]
Integrate both sides:
[ ln sqrt{v^2 - 1} pm ln x - ln C ]
Back Substitution: Replace (v) with (frac{y}{x}) and simplify.
[ ln sqrt{left( frac{y}{x} right)^2 - 1} pm ln x - ln C ]
Exponentiate both sides:
[ sqrt{left( frac{y}{x} right)^2 - 1} Cx^{-1} Big( text{or} ) Cx ]
Finally, solve for (y):
[ y^2 - x^2 (Cx Big)^2 ] or [ sqrt{y^2 - x^2} pm Cx ]
The general solution can be written as:
[ y pm sqrt{C^2x^2 x^2} ] or [ sqrt{y^2 - x^2} pm C ]
Conclusion
In conclusion, the general solution to the given first-order differential equation with degree 2 is:
[ y pm sqrt{C^2x^2 x^2} ] or [ sqrt{y^2 - x^2} pm C ]
This solution shows how the process of solving a differential equation can be systematic and logical, especially using substitution and integration techniques.