Question

Question is here in the photo

Answer 1

`f(x)=6e^{3x^(18)}`

Explanation:

we can separate the variables in the given equation and the integrate respect to x variable:

`(dy)/(dx)=54yx^(17)\\\\(1)/(y)(dy)/(dx)=54x^(17)\\\int\ {(1)/(y)(dy)/(dx)} \, dx = \int\ {54x^(17)} \, dx\\\int\ {(1)/(y)} \, dy = \int\ {54x^(17)} \, dx\\\\ ln|y|=(54)/(17+1)x^(17+1)+C\\ln|y|=3x^(18)+C\\y=e^{3x^(18)+C}\\y=ke^{3x^(18)}`

where
`k = e^C\\`

finally, as
`f(0)=6` then:

`6=ke^{0^(18)}=ke^(0)=k(1)=k`