I have the following differential equation >> dx/dt = -1/2 (x)

it says i need to verify that x(t)=Ce^-t/2

i have done:

rearrange >> 1/x dx = -1/2 dt

integrate >> ln x = -t/2 + C

take log to simplify >> ln x = ln e^-t/2 + ln C

>> ln x = ln (Ce^-t/2)

which finally gives >> x = Ce^-t/2

is this correct or am i doing it wrong??

It then goes on to say find the particular solution that satisfies x(0) = 2 and sketch on a direction field...so:

input the condition >> 2 = Ce^-0/2

simplify >> 2 = Ce^0

>> 2 = C(1)

therefore >> 2 = C

and the particular solution is >> x = 2e^-t/2

again is this correct? And how would i go about sketching this?

Thanks