Question

Find the solution to the differential equation \[\frac{ dy }{ dx }+\frac{ y }{ 5 }=0 \]

with the initial condition y(0)=8

Answer 1

So, ive done separation of variables, and integrated dy/(-y) amd dx/5 and gotten -ln(y) = x/5+C, but I am not sure what to do next.

Answer 2

try to isolate y

Answer 3

if ln a= b
then a=e^b

Answer 4

when x=0 y=8
plug that solve for c and yeah isolate y

Answer 5

ok, so \[-\ln(y)=\frac{ x }{ 5 }+C\] becomes \[-y=e^(\frac{ x }{ 5 }+C)\]

Answer 6

so -8=e^(0+c), -8=e^c?

Answer 7

c=-ln(8) ?

Answer 8

yup, c=-ln 8
so
-ln y +ln 8 = x/5
ln (y/8) = -x/5
got this ?

Answer 9

property of log,
log A-log B =log (A/B)

Answer 10

ahhhh yes I see that

Answer 11

y=8*e^(x/5)

Answer 12

y=8*e^(-x/5)

Answer 13

Haha yes, making every mistake posssible on this one. Thanks for the help!

Answer 14

welcome ^_^