Eliminate Constants 1Bd350

1. **State the problem:** We are given the function $$y = C_1 x + C_2 x^2$$ with arbitrary constants $$C_1$$ and $$C_2$$. We want to eliminate these constants to find a differential equation involving only $$x$$, $$y$$, and derivatives of $$y$$. 2. **Given derivatives:** $$y' = C_1 + 2 C_2 x$$ $$y'' = 2 C_2$$ 3. **Express constants in terms of derivatives:** From $$y'' = 2 C_2$$, isolate $$C_2$$: $$C_2 = \frac{y''}{2}$$ From $$y' = C_1 + 2 C_2 x$$, substitute $$C_2$$: $$C_1 = y' - 2 C_2 x = y' - 2 \cdot \frac{y''}{2} x = y' - y'' x$$ 4. **Substitute $$C_1$$ and $$C_2$$ back into $$y$$:** $$y = C_1 x + C_2 x^2 = (y' - y'' x) x + \frac{y''}{2} x^2 = y' x - y'' x^2 + \frac{y''}{2} x^2$$ Simplify the terms involving $$y''$$: $$- y'' x^2 + \frac{y''}{2} x^2 = - \frac{y''}{2} x^2$$ So, $$y = y' x - \frac{y''}{2} x^2$$ 5. **Rearrange to form the differential equation:** Bring all terms to one side: $$y - y' x + \frac{y''}{2} x^2 = 0$$ Multiply both sides by 2 to clear the fraction: $$2 y - 2 y' x + y'' x^2 = 0$$ 6. **Final differential equation:** $$\boxed{y'' x^2 - 2 x y' + 2 y = 0}$$ This is the differential equation with the arbitrary constants eliminated. 7. **Verification of the derivative expression:** Given the expression $$\frac{d}{dx} (x y' - y)$$, compute it: $$\frac{d}{dx} (x y' - y) = y' + x y'' - y' = x y''$$ This matches the derivative of the expression inside the parentheses. **Summary:** - We isolated $$C_1$$ and $$C_2$$ using $$y'$$ and $$y''$$. - Substituted back into $$y$$ to eliminate constants. - Derived the differential equation $$y'' x^2 - 2 x y' + 2 y = 0$$. This completes the elimination by the isolation method.