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?