Questions: y'(sinh 3y - 2xy) = y^2

Solution Steps

To solve the given differential equation \( y'(\sinh(3y) - 2xy) = y^2 \), we can follow these steps:

1. Recognize that this is a first-order differential equation.
2. Rewrite the equation in a more standard form if possible.
3. Use numerical methods to solve the differential equation, as it may not have a straightforward analytical solution.

Given the differential equation: \[ y'(\sinh(3y) - 2xy) = y^2 \]

We can rewrite \( y' \) as \( \frac{dy}{dx} \): \[ \frac{dy}{dx} (\sinh(3y) - 2xy) = y^2 \]

To solve this differential equation, we need to separate the variables \( x \) and \( y \). We can do this by dividing both sides by \( y^2 (\sinh(3y) - 2xy) \): \[ \frac{1}{y^2} \, dy = \frac{dx}{\sinh(3y) - 2xy} \]

Now, we integrate both sides of the equation. The left side with respect to \( y \) and the right side with respect to \( x \): \[ \int \frac{1}{y^2} \, dy = \int \frac{dx}{\sinh(3y) - 2xy} \]

The left side integral is straightforward: \[ \int y^{-2} \, dy = -y^{-1} = -\frac{1}{y} \]

The right side integral is more complex and depends on the specific form of \( \sinh(3y) - 2xy \). However, without additional context or simplification, we cannot integrate it directly.

Final Answer