4 min read Jun 17, 2024

Solving the Differential Equation: (x^2 + 2y^2) dx/dy = xy

This article will explore the solution to the given differential equation: (x^2 + 2y^2) dx/dy = xy. We will use a combination of techniques to arrive at the general solution.

1. Recognizing the Type of Differential Equation

The equation is a first-order, non-linear differential equation. This means it involves the first derivative of the dependent variable (y) and possibly other terms involving both x and y, but not higher-order derivatives.

2. Rearranging the Equation

First, let's rearrange the equation to make it easier to work with:

dx/dy = (xy) / (x^2 + 2y^2)

Now, we can see that the equation is in the form of a homogeneous differential equation. This means the right-hand side can be expressed as a function of (y/x) only.

3. Substitution and Integration

To solve the homogeneous equation, we use the substitution v = y/x. This implies y = vx, and differentiating both sides with respect to x gives us:

dy/dx = v + x dv/dx

Substituting these expressions into our rearranged equation:

v + x dv/dx = (x * vx) / (x^2 + 2(vx)^2)

Simplifying the equation:

v + x dv/dx = v / (1 + 2v^2)

Now, we can separate variables and integrate:

x dv/dx = v / (1 + 2v^2) - v
x dv/dx = -v^3 / (1 + 2v^2)
(1 + 2v^2) / v^3 dv = -dx / x

Integrating both sides:

∫(1/v^3 + 2/v) dv = -∫dx / x
-1/(2v^2) + 2ln|v| = -ln|x| + C

Where C is the constant of integration.

4. Back Substitution

Now, we substitute back v = y/x:

-1/(2(y/x)^2) + 2ln|y/x| = -ln|x| + C

Simplifying the equation:

-x^2 / (2y^2) + 2ln|y| - 2ln|x| = -ln|x| + C
-x^2 / (2y^2) + 2ln|y| - ln|x| = C

5. The General Solution

Therefore, the general solution to the differential equation (x^2 + 2y^2) dx/dy = xy is:

-x^2 / (2y^2) + 2ln|y| - ln|x| = C

where C is an arbitrary constant.


This solution represents a family of curves satisfying the given differential equation. Each curve corresponds to a specific value of the constant C.