If a curve passes through the origin and the slope of the tangent at (x,y) is x^2-4x+y+8x-…

Question

If a curve passes through the origin and the slope of the tangent at $(x,y)$ is $\dfrac{x^2-4x+y+8}{x-2}$, then this curve also passes through:

Accepted Answer

With $t=x-2,u=y+4$: $\dfrac{du}{dt}=t+\dfrac{u}{t} \implies \dfrac{u}{t}=t+c$.

Through origin: $\dfrac{4}{-2}=-2+c \Rightarrow c=0$. So $y+4=(x-2)^2$, which is satisfied by $(5,5)$: $9=9$.

Solutions · 1