1. **Stating the problem:** Solve the differential equation: $$x^{2} (y + a)^{2} (y' - 1) = y^{2} - 2ax^{2} y + a^{2}$$ where $y' = \frac{dy}{dx}$. 2. **Rewrite the equation:** Expand the right side and isolate $y'$: $$x^{2} (y + a)^{2} (y' - 1) = y^{2} - 2ax^{2} y + a^{2}$$ Divide both sides by $x^{2} (y + a)^{2}$: $$y' - 1 = \frac{y^{2} - 2ax^{2} y + a^{2}}{x^{2} (y + a)^{2}}$$ 3. **Simplify the numerator:** Notice that: $$y^{2} - 2ax^{2} y + a^{2} = (y - a x)^{2}$$ Check: $$(y - a x)^{2} = y^{2} - 2 a x y + a^{2} x^{2}$$ But the original numerator is $y^{2} - 2 a x^{2} y + a^{2}$, which is not the same. So this factorization is incorrect. Instead, keep numerator as is. 4. **Rewrite $y'$:** $$y' = 1 + \frac{y^{2} - 2 a x^{2} y + a^{2}}{x^{2} (y + a)^{2}}$$ 5. **Try substitution:** Let $z = y + a$, then $y = z - a$ and $y' = z'$. Rewrite numerator: $$y^{2} - 2 a x^{2} y + a^{2} = (z - a)^{2} - 2 a x^{2} (z - a) + a^{2}$$ Expand: $$(z - a)^{2} = z^{2} - 2 a z + a^{2}$$ So numerator becomes: $$z^{2} - 2 a z + a^{2} - 2 a x^{2} z + 2 a^{2} x^{2} + a^{2}$$ Combine like terms: $$z^{2} - 2 a z - 2 a x^{2} z + a^{2} + 2 a^{2} x^{2} + a^{2} = z^{2} - 2 a z (1 + x^{2}) + a^{2} (1 + 2 x^{2})$$ 6. **Rewrite denominator:** $$x^{2} (y + a)^{2} = x^{2} z^{2}$$ 7. **Rewrite $y'$ in terms of $z$:** $$z' = 1 + \frac{z^{2} - 2 a z (1 + x^{2}) + a^{2} (1 + 2 x^{2})}{x^{2} z^{2}}$$ 8. **Separate terms:** $$z' = 1 + \frac{z^{2}}{x^{2} z^{2}} - \frac{2 a z (1 + x^{2})}{x^{2} z^{2}} + \frac{a^{2} (1 + 2 x^{2})}{x^{2} z^{2}}$$ Simplify: $$z' = 1 + \frac{1}{x^{2}} - \frac{2 a (1 + x^{2})}{x^{2} z} + \frac{a^{2} (1 + 2 x^{2})}{x^{2} z^{2}}$$ 9. **Final form:** $$z' = 1 + \frac{1}{x^{2}} - \frac{2 a (1 + x^{2})}{x^{2} z} + \frac{a^{2} (1 + 2 x^{2})}{x^{2} z^{2}}$$ This is a nonlinear first order ODE in $z$. Further solution would require advanced methods or numerical approaches. --- **Summary:** We transformed the original equation into an ODE for $z = y + a$: $$z' = 1 + \frac{1}{x^{2}} - \frac{2 a (1 + x^{2})}{x^{2} z} + \frac{a^{2} (1 + 2 x^{2})}{x^{2} z^{2}}$$ This form is suitable for further analysis or numerical solution.