Which of the following equation is a homogeneous differential equation?

Question

Accepted Answer

To determine which equation is a homogeneous differential equation, we need to understand what a homogeneous differential equation is. A differential equation is considered homogeneous if it can be expressed in the form where all terms are of the same degree when considering the variables involved.

Let's analyze each option:

1. **Option 1: $(4x+6y+5)dy-(3y+2x+4)dx=0$**

- The terms in this equation are not homogeneous because they do not have the same degree. For instance, the term $4x$ is of degree 1, $6y$ is of degree 1, but the constant term $5$ is of degree 0. Similarly, $3y$ and $2x$ are of degree 1, but the constant $4$ is of degree 0. Therefore, this equation is not homogeneous.

2. **Option 2: $(xy)dx-(x^{3}+y^{3})dy=0$**

- In this equation, the term $xy$ is of degree 2 (since $x$ and $y$ are each of degree 1), while $x^3$ and $y^3$ are of degree 3. The degrees of the terms are not consistent, so this equation is not homogeneous.

3. **Option 3: $(x^{3}+2y^{2})dx+2xy~dy=0$**

- Here, $x^3$ is of degree 3, $2y^2$ is of degree 2, and $2xy$ is of degree 2. The terms do not all have the same degree, so this equation is not homogeneous.

4. **Option 4: $y^{2}dx+(x^{2}-xy-y^{2})dy=0$**

- In this equation, $y^2$ is of degree 2, $x^2$ is of degree 2, $xy$ is of degree 2, and $y^2$ is of degree 2. All terms are of degree 2, which means this equation is homogeneous.

Therefore, Option 4 is the correct answer because it is the only equation where all terms are of the same degree, satisfying the condition for a homogeneous differential equation.