Family of curves

Question

Find the DE of straight lines with slope twice the x-intercept

Answer 1

Let we have y = mx + b. So m = 2b. Equation y = 2bx + b. From here y' = 2b.

So we have: y = y'x + y'/2. y' = y/(x + 1/2);

dy/dx = y/(x + 1/2) is differential equation of family of lines with sope 2 times y-intercept.

Sorry, I solved when slope 2 times y-intercept.

Now if y = mx + b. y = 0 x = - b/m.

So m = - 2b/m, m² = - 2b. Now m = y', b = - (y')²/2.

So ODE looking like this: y = y'x - (y')²/2 or (y')² - 2x y' + 2y = 0, or y' = x ± (x² - 2y)^1/2