Question

Question in attachment.

Answer 1

The general solution of equations of the form
`g'(x)=kg(x)` is
`g(x)=C*e^k^x` for some constant C.

This can be found using separation of variables.


`(dg)/(dx) = kg`


`(dg)/(g)=kdx`


`\int\limits (dg)/(g) = \int\limits kdx`


`ln(IgI)=kx+c`


`e^l^n^(^I^g^I^)=e^(kx+c)`


`g=c\cdot e^k^x`
`Let \ C=e^c \geq 0`

In our case
`k=8` so
`g(x)=C\cdot e^(8x)`

Let's use the fact that
`g(2)=7` to find C.


`g(x)=c\cdot e^8^x`


`g(2)=c\cdot e^8^.^2`
`Plug \ x=2`


`7=c\cdot e^8^.^2`
`g(2)=7`


`7e^-^1^6 = c`

In conclusion,
`g(x)=7e^(8x-16)`.