Question

Solve: \[ x^2 \, dy - (3x^2 + xy + y^2) \, dx = 0 \quad \text{given that} \quad y = 1 \, \text{when} \, x = 1 \]

Hint:

For non-separable first-order differential equations, consider using the method of integrating factors or substitution to find a solution.

Answer 1

Step 1: Rearranging the equation. 
We can rearrange the given equation as follows: \[ x^2 \, dy = (3x^2 + xy + y^2) \, dx \] Step 2: Solving the equation. 
This is a first-order differential equation that can be solved using separation of variables or another appropriate method. First, we attempt to separate the variables and integrate both sides. Alternatively, you may need to use an integrating factor if the equation is linear. After solving the equation and using the initial condition \( y = 1 \) when \( x = 1 \), we can find the particular solution. 
Step 3: Conclusion. 
The solution to the equation involves advanced integration techniques, and the particular solution can be found by using the initial condition.