Solving Differential Equations: Techniques and Examples

Differential equations are fundamental to the study of many physical and mathematical systems. This article will explore various techniques to solve differential equations, focusing on how to solve two specific problems: y^2 - 2xy dx x^2 - 2xy dy and dy/dx y^2 - 2xy / x^2 - 2xy. Each example will be approached using different methods, including exact equations and integrating factors.

Solving the Differential Equation: y^2 - 2xy dx x^2 - 2xy dy

The given differential equation is:

"y^2 - 2xy dx x^2 - 2xy dy"

To solve it, we can start by rewriting it in a more manageable form:

y^2 - 2xy dx - x^2 - 2xy dy 0

This is a first-order differential equation that can be approached using the method of exact equations or by finding an integrating factor if it is not exact.

Step 1: Check for Exactness

We first identify the functions: M(x, y) y^2 - 2xy and N(x, y) -x^2 - 2xy. The next step is to calculate the partial derivatives:

?M/?y 2y - 2x

?N/?x -2x - 2y

Since ?M/?y ?N/?x, the equation is exact.

Step 2: Find the Potential Function

To find a potential function Ψ(x, y) such that dΨ M dx - N dy, we integrate M with respect to x:

Ψ(x, y) ∫(y^2 - 2xy) dx y^2x - x^2y h(y)

Here, h(y) is an arbitrary function of y. Next, we differentiate Ψ with respect to y:

?Ψ/?y 2xy - x^2 h'(y)

We set this equal to N:

2xy - x^2 h'(y) -x^2 - 2xy

This simplifies to:

2xy - x^2 h'(y) -x^2 - 2xy

From this equation, we can see that h'(y) -4xy, which implies that h(y) 0. Therefore, the potential function simplifies to:

Ψ(x, y) y^2x - x^2y C

where C is a constant.

Step 3: The Implicit Solution

The implicit solution to the differential equation is:

y^2x - x^2y C

This describes a family of curves in the xy-plane depending on the value of the constant C.

Alternative Approach: Simplifying with Substitution

Given the differential equation dy/dx y^2 - 2xy / x^2 - 2xy, we can simplify it using the substitution yx xv. Let u y / x. This gives:

dy/dx x dv/dx v

The differential equation then becomes:

x dv/dx v u^2 - 2uv / x^2 - 2uv

Simplifying, we get:

dv/dx (3u^2 - u - 2v) / x

Letting v u / h, we can further simplify to:

dv/dx -3u^2 - u / x(x - 2u)

This can be solved by separation of variables:

log(u - u^2) -3 log(x) log(c)

Which simplifies to:

x^2y - y^2x C

This is the required general solution.