Solving Differential Equations with Nonlinear and Higher-Degree Variables

Understanding and solving differential equations (DEs) with nonlinear and higher-degree variables is a crucial skill in mathematics and its applications. This article provides a detailed guide on how to solve an interesting differential equation of the form y xy' x^2 / y' 1, demonstrating various techniques used in the process.

Introduction to the Problem

The given differential equation is:

y xy' x^2 / y' 1

This equation can be rewritten by letting y' p. Substituting this, we get:

y xp x^2 / p 1

Step-by-Step Solution

To proceed, we differentiate the equation with respect to x.

dy/dx p x(dp/dx) (2x/p - x^2/p^2 * dp/dx)

Since dy/dx p, we substitute this into the equation, resulting in:

p p x(dp/dx) (2x/p - x^2/p^2 * dp/dx)

After simplifying, we get the following linear DE:

dp/dx - x/(2p) -p/2

The integrating factor (IF) for this linear DE is:

IF p^(-1/2)

Multiplying both sides by this integrating factor, we get:

p^(-1/2) * dp/dx - (x/2) * p^(-3/2) -p^(-1/2)

Integrating both sides, we have:

x -1/3 * p^(3/2) * C

Substituting this back into the original equation, we obtain:

y (C * p^(1/2) - 1/3 * p^2)^2 / p

Conclusion

The solution to the differential equation is:

y (C * p^(1/2) - 1/3 * p^2)^2 / p

This approach effectively combines methods of substitution and integration to solve nontrivial differential equations. The key steps involve transforming the equation into a more manageable form, applying integration techniques, and substituting back to find the final solution.

Understanding these methods not only solves the given problem but also provides a robust framework for tackling similar nonlinear and higher-degree differential equations in various mathematical and scientific contexts.