(x^2+xy+4x+2y+4)dy/dx=y^2

4 min read Jun 17, 2024

Solving the Differential Equation: (x^2+xy+4x+2y+4)dy/dx = y^2

This article will guide you through solving the differential equation (x^2+xy+4x+2y+4)dy/dx = y^2. This is a first-order, non-linear, and separable differential equation. Let's break down the solution process step-by-step.

1. Separating Variables

The first step is to separate the variables (x and y) on each side of the equation. We can do this by rearranging the terms:

(x^2+xy+4x+2y+4)dy = y^2 dx

Next, we aim to get all the 'y' terms on the left-hand side and all the 'x' terms on the right-hand side:

dy / y^2 = dx / (x^2+xy+4x+2y+4)

2. Simplifying the Right-Hand Side

The right-hand side looks complex. We can simplify it by factoring the denominator:

dy / y^2 = dx / [(x+2)(x+y+2)]

Now the equation is in a form where we can integrate both sides.

3. Integration

Integrate both sides of the equation:

∫ dy / y^2 = ∫ dx / [(x+2)(x+y+2)]

The left-hand side is a simple integral:

-1/y = ∫ dx / [(x+2)(x+y+2)]

The right-hand side requires a technique called partial fraction decomposition. This involves expressing the integrand as a sum of simpler fractions:

1/[(x+2)(x+y+2)] = A/(x+2) + B/(x+y+2)

Solving for A and B, we find:

A = 1/(y-2) 
B = -1/(y-2)

Substituting these back into the integral:

∫ dx / [(x+2)(x+y+2)] = ∫ [1/(y-2)]/(x+2) dx - ∫ [1/(y-2)]/(x+y+2) dx

Now we can integrate each term:

∫ dx / [(x+2)(x+y+2)] = [1/(y-2)] * ln|x+2| - [1/(y-2)] * ln|x+y+2| + C

Where C is the constant of integration.

4. Combining Results and Solving for y

Substituting the results of the integration back into the original equation:

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

Simplifying and solving for y will provide the general solution to the differential equation.

Note: This solution may involve implicit expressions of y in terms of x and the constant of integration, making it difficult to explicitly solve for y. You may need to use numerical methods or graphical techniques to analyze the solution for specific values of the constant of integration.