To solve the differential equation
\[ y' = \frac{xy}{x^2} - y^2, \]
we can first rewrite the equation for clarity:
\[ y' = \frac{y}{x} - y^2. \]
This is a first-order nonlinear differential equation. We can rearrange it to a more standard form:
\[ y' + y^2 = \frac{y}{x}. \]
Now, let's reorganize it one more time:
\[ y' = \frac{y}{x} - y^2. \]
This kind of equation can often be approached using the technique of separation of variables. However, it may be simpler to recognize that it may be solvable through substitution or Bernoulli's methods. Let's transform the equation into a separable form.
First, we can move terms involving \( y \) to one side:
\[ \frac{dy}{dx} + y^2 = \frac{y}{x}. \]
This can also be expressed in a form that allows for substitution by dividing through by \( y^2 \):
\[ \frac{dy}{dx} + y^2 = \frac{y}{x}. \]
Rearranging gives:
\[ \frac{dy}{dx} = -y^2 + \frac{y}{x}. \]
The equation is separable:
\[ \frac{dy}{y(-y + \frac{1}{x})} = dx. \]
Now we can integrate both sides. The left-hand side requires partial fraction decomposition where:
\[ \frac{1}{y(-y + \frac{1}{x})} = \frac{A}{y} + \frac{B}{-y + \frac{1}{x}}. \]
Finding constants \( A \) and \( B \), we will multiply out and solve for equality. After finding a common denominator and rearranging, we integrate both sides. The integration yields:
- For \(\frac{1}{y}\), we have:
\[ \int \frac{1}{y} dy = \ln |y|, \]
- For \(-y + \frac{1}{x}\), we have:
\[ -\ln |y - \frac{1}{x}|. \]
Combining results and exponentiating gives implicit solutions, which can be written in general form depending on the integration constant:
Concisely, results yield solutions for \(y\) in terms of natural logs through transformations and the methods outlined. Additional manipulations will refine the expression of terms to reflect desired accuracy levels based on convergence under varying boundary conditions, which can influence constants of integration \(C\) derived from integral evaluations.
Overall, decomposing adequately to extract primal values leads towards solution yielding:
\[ C = \ln\left|y \right| + \frac{1}{x}. \]
Finally, we conclude with implicit differential equation solutions representing dynamic interactions encapsulated within contextually relevant parameters, however these can be expressed through direct graphical analysis or distinct numerical simulations tailored to observational data acquisition and interpolative validations focusing directly on predominant measures for \( y \) based on initial conditions provided upon evaluations relative to \( C\).
Thus, the approach for determining \(y\) remains fluid dependent upon variable shifts and analytic methodologies adopted, but necessitates contextual apprenticeships aligning alongside core mathematical principles applied in problem-solving.
3 answers