Question

Determine the region of the xy-plane where the solution of the DE would have a unique solution passing through the point (x0,y0).
(9-y^2)dy/dx=x^4

Answer 1

Answer:

given equation,

$(9-y^2)\dfrac{dy}{dx}= x^4\\(9-y^2)dy = x^4 dx\\\int (9-y^2)dy =\int x^4 dx\\9y -\dfrac{y^3}{3} = \dfrac{x^5}{5}+C$

so,

the given point from where the region is passing through is (x₀, y₀)

hence, the unique equation comes out to be

$9y_0 -\dfrac{y_0^3}{3} = \dfrac{x_0^5}{5}+C$

$C = 9y_0 -\dfrac{y_0^3}{3}-\dfrac{x_0^5}{5}$

hence unique equation,

$9y -\dfrac{y^3}{3} = \dfrac{x^5}{5}+ 9y_0 -\dfrac{y_0^3}{3}-\dfrac{x_0^5}{5}$