[latex]
\[
\textbf{Q1: Solve the differential equation:} \quad x^2 y + 1 \, dx + y^2 x + 1 \, dy = 0
\]
\[
\textbf{Step 1: Rearranging the equation}
\]
We start by separating the terms involving \( dx \) and \( dy \). Rewrite the equation as:
\[
(x^2 y + 1) \, dx = -(y^2 x + 1) \, dy
\]
\[
\textbf{Step 2: Separation of variables}
\]
Now, we try to separate the variables \( x \) and \( y \). To do this, divide both sides by \( (x^2 y + 1)(y^2 x + 1) \):
\[
\frac{dx}{y^2 x + 1} = -\frac{dy}{x^2 y + 1}
\]
\[
\textbf{Step 3: Integration of both sides}
\]
We now integrate both sides. The left-hand side integral is:
\[
\int \frac{dx}{y^2 x + 1}
\]
The right-hand side integral is:
\[
\int -\frac{dy}{x^2 y + 1}
\]
The integrals on both sides are not trivial, and we may need to apply techniques such as substitution or partial fraction decomposition. However, solving these integrals directly may not yield an elementary function. Therefore, we consider two possible cases for this equation:
### Case 1: Numerical Methods
In this case, solving the equation numerically using software such as MATLAB, Mathematica, or Wolfram Alpha is the best approach. You would use the software to solve for the integral values of \( x \) and \( y \) given initial conditions.
### Case 2: Special Functions or Approximation
Sometimes, for non-trivial integrals, solutions can be expressed in terms of special functions or using approximation techniques such as series expansion. If we find the integrals involve standard forms, solutions might involve functions like the logarithm, arctangent, or other common special functions.
\[
\textbf{Step 4: General Solution}
\]
If we perform the integration, the general solution might involve implicit functions or relations. The final solution could be expressed as:
\[
F(x, y) = C
\]
where \( C \) is a constant of integration, and \( F(x, y) \) represents the implicit relationship between \( x \) and \( y \).
Thus, the solution to this equation can vary based on the method used for solving the integrals, which could be expressed numerically or in terms of special functions.