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

given equation,

`(9-y^2)(dy)/(dx)= x^4\\(9-y^2)dy = x^4 dx\\\int (9-y^2)dy =\int x^4 dx\\9y -(y^3)/(3) = (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 -(y_0^3)/(3) = (x_0^5)/(5)+C`

`C = 9y_0 -(y_0^3)/(3)-(x_0^5)/(5)`

hence unique equation,

`9y -(y^3)/(3) = (x^5)/(5)+ 9y_0 -(y_0^3)/(3)-(x_0^5)/(5)`