To solve the differential equation
\[ xy'^2 + 2xy' - y = 0, \]
we can denote \(y' = \frac{dy}{dx}\). We will replace \(y'\) with \(p\), giving us:
\[ xp^2 + 2xp - y = 0. \]
This is a quadratic equation in \(p\):
\[ xp^2 + 2xp - y = 0. \]
We can apply the quadratic formula \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = x\), \(b = 2x\), and \(c = -y\):
\[ p = \frac{-2x \pm \sqrt{(2x)^2 - 4x(-y)}}{2x}, \] \[ p = \frac{-2x \pm \sqrt{4x^2 + 4xy}}{2x}, \] \[ p = \frac{-2x \pm 2\sqrt{x^2 + xy}}{2x}. \]
Simplifying, we have:
\[ p = -1 \pm \sqrt{1 + \frac{y}{x}}. \]
This gives us two possible expressions for \(p\):
- \(p = -1 + \sqrt{1 + \frac{y}{x}}\)
- \(p = -1 - \sqrt{1 + \frac{y}{x}}\)
We can focus on the first case:
\[ \frac{dy}{dx} = -1 + \sqrt{1 + \frac{y}{x}}. \]
Now, we can separate the variables:
\[ dy = \left(-1 + \sqrt{1 + \frac{y}{x}}\right) dx. \]
To find a solution, we can analyze this further. We will rewrite this equation into a more separable form or use appropriate substitutions if necessary.
For the second case:
\[ \frac{dy}{dx} = -1 - \sqrt{1 + \frac{y}{x}}. \]
We can follow a similar approach with this equation.
To summarize, we need to integrate the expressions to find \(y\) in terms of \(x\):
- For the first case, integrate \(dy\).
- For the second case, also integrate \(dy\).
I'll proceed with the first case:
\[ \frac{dy}{dx} = -1 + \sqrt{1 + \frac{y}{x}}. \]
This is a more complex equation to integrate directly, so let's denote the two results for the two cases:
- \(y_1 = x(-1 + \sqrt{1 + \frac{y_1}{x}})\)
- \(y_2 = x(-1 - \sqrt{1 + \frac{y_2}{x}})\)
Each can give different families of solutions depending on initial conditions or constraints. Further elaboration or numerical methods might be necessary to approach specific solutions based on context.
Thus, we obtain two forms of \(y\) based on the differential equation. Further simplification might depend on specific boundary conditions or integrate these expressions more explicitly for practical application. Would you like specific assistance regarding one of these results further?
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