Question

Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]

Hint:

For separable equations, always separate \( x \) and \( y \) terms before integrating and simplify using logarithmic or exponential properties as needed.

Answer 1

Step 1: Rewrite the equation.
The given equation is: \[ y \, dx = (x + 2y^2) \, dy. \] Rearrange terms to separate \( x \) and \( y \): \[ \frac{dx}{x} = \frac{1}{y} \, dy + 2y \, dy. \] Step 2: Integrate both sides.
- For the left-hand side: \[ \int \frac{dx}{x} = \ln|x| + C_1, \] where \( C_1 \) is the constant of integration. - For the right-hand side, split into two integrals: \[ \int \frac{1}{y} \, dy + \int 2y \, dy. \] - First term: \[ \int \frac{1}{y} \, dy = \ln|y|. \] - Second term: \[ \int 2y \, dy = y^2. \] Thus: \[ \int \frac{1}{y} \, dy + \int 2y \, dy = \ln|y| + y^2. \] Step 3: Combine results.
Equating the results: \[ \ln|x| = \ln|y| + y^2 + C_1. \] Step 4: Simplify the equation.
Let \( C = -C_1 \) (a constant), then: \[ \ln|x| - \ln|y| = y^2 + C. \] Using the logarithmic property \( \ln|x| - \ln|y| = \ln\left|\frac{x}{y}\right| \), we get: \[ \ln\left|\frac{x}{y}\right| = y^2 + C. \] Step 5: Write the general solution.
Exponentiate both sides to simplify: \[ \frac{x}{y} = e^{y^2 + C} = e^C \cdot e^{y^2}. \] Let \( e^C = K \), where \( K \) is a constant, so: \[ \frac{x}{y} = K e^{y^2}. \] Finally: \[ x = Ky e^{y^2}. \] Conclusion:
The general solution is: \[ \boxed{x = Ky e^{y^2}}. \]