The general solution of equations of the form [tex]g'(x)=kg(x)[/tex] is [tex]g(x)=C*e^k^x[/tex] for some constant C.
This can be found using separation of variables.
[tex] \dfrac{dg}{dx} = kg [/tex]
[tex] \dfrac{dg}{g}=kdx [/tex]
[tex] \int\limits \dfrac{dg}{g} = \int\limits kdx[/tex]
[tex]ln(IgI)=kx+c[/tex]
[tex]e^l^n^(^I^g^I^)=e^{kx+c}[/tex]
[tex]g=c\cdot e^k^x[/tex] [tex]Let \ C=e^c \geq 0[/tex]
In our case [tex]k=8[/tex] so [tex]g(x)=C\cdot e^{8x}[/tex]
Let's use the fact that [tex]g(2)=7[/tex] to find C.
[tex]g(x)=c\cdot e^8^x[/tex]
[tex]g(2)=c\cdot e^8^.^2[/tex] [tex]Plug \ x=2[/tex]
[tex]7=c\cdot e^8^.^2[/tex] [tex]g(2)=7[/tex]
[tex]7e^-^1^6 = c[/tex]
In conclusion, [tex]g(x)=7e^{8x-16}[/tex].
