x² y² dx 2xydy 0
[latex]
\[
\textbf{Q: Solve the differential equation:} \quad x^2 y^2 \, dx – 2xy \, dy = 0
\]
\[
\textbf{Step 1: Rearrange the equation}
\]
Rearrange the equation to separate variables:
\[
x^2 y^2 \, dx = 2xy \, dy
\]
\[
\textbf{Step 2: Separate the variables}
\]
We can now separate the variables \(x\) and \(y\):
\[
\frac{x^2}{xy} \, dx = \frac{2}{y} \, dy
\]
Simplifying the terms:
\[
\frac{x}{y} \, dx = \frac{2}{y} \, dy
\]
\[
\textbf{Step 3: Integrate both sides}
\]
Now, integrate both sides:
\[
\int \frac{x}{y} \, dx = \int \frac{2}{y} \, dy
\]
On integrating:
\[
\frac{x^2}{2} = 2 \ln |y| + C
\]
\[
\textbf{Step 4: Solve for y}
\]
To solve for \(y\), first isolate \( \ln |y| \):
\[
\ln |y| = \frac{x^2}{4} + C_1
\]
Exponentiate both sides:
\[
|y| = e^{\frac{x^2}{4} + C_1}
\]
Since \( e^{C_1} \) is a constant, we can write it as a new constant \( C_2 \):
\[
|y| = C_2 e^{\frac{x^2}{4}}
\]
Thus, the general solution is:
\[
y = C e^{\frac{x^2}{4}}
\]
where \(C\) is a constant.
\[
\textbf{Final Solution:}
\]
\[
y = C e^{\frac{x^2}{4}}
\]