(xy-2y^2)dx-(x^2-3xy)dy=0

Jasonbradley

4 min read Jun 17, 2024

Solving the Differential Equation: (xy - 2y^2)dx - (x^2 - 3xy)dy = 0

This article will guide you through the process of solving the given differential equation:

(xy - 2y^2)dx - (x^2 - 3xy)dy = 0

This equation is a homogeneous differential equation because we can rewrite it in the form:

M(x, y)dx + N(x, y)dy = 0

where M(x, y) and N(x, y) are homogeneous functions of the same degree. In our case, both M(x, y) = xy - 2y^2 and N(x, y) = -x^2 + 3xy are homogeneous functions of degree 2.

Here's how to solve the equation:

1. Substitution and Simplification

We can solve homogeneous differential equations using the substitution y = vx. This gives us:

dy = vdx + xdv

Substituting these into our original equation, we get:

(x(vx) - 2(vx)^2)dx - (x^2 - 3x(vx))(vdx + xdv) = 0

Simplifying the equation:

(vx^2 - 2v^2x^2)dx - (x^2 - 3vx^2)(vdx + xdv) = 0

2. Separating Variables

Now, we can separate the variables 'x' and 'v':

(vx^2 - 2v^2x^2 - x^2v + 3v^2x^2)dx = (x^2 - 3vx^2)xdv

Combining like terms and factoring:

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

Dividing both sides by x^2(1 - 3v)(v + 2v^2):

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

3. Integrating Both Sides

Now we can integrate both sides of the equation:

∫(1/x)dx = ∫((1 - 3v) / (v + 2v^2))dv

The left side integrates to ln|x|. For the right side, we need to use partial fractions.

4. Partial Fractions

Decomposing the right side:

(1 - 3v) / (v + 2v^2) = A / v + B / (1 + 2v)

Solving for A and B, we get:

A = 1
B = -4

Now the integral becomes:

∫(1/x)dx = ∫(1/v)dv - ∫(4 / (1 + 2v))dv

5. Solving the Integral

Integrating both sides:

ln|x| = ln|v| - 2ln|1 + 2v| + C

where C is the constant of integration.

6. Back Substitution and Solution

Substituting back v = y/x, we get:

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

Simplifying:

ln|x| = ln|y| - ln|x| - 2ln|x + 2y| + C

Combining logarithms and exponentiating both sides:

x^2 = Ce^(y/(x + 2y))

This is the general solution to the given differential equation.

7. Final Notes

Remember that the constant of integration 'C' can be any real number.
We can find a particular solution for a given initial condition.
The solution might not be defined for all values of x and y.

This article provides a comprehensive guide to solving the given differential equation. However, always remember to double-check your calculations and use appropriate methods for each step.