# Math Help - differentiation, separation of variables???

1. ## differentiation, separation of variables???

the positive quantities x,y,z are related and vary with time t, where t is bigger or equal to 0. The value of x is described by the differential equation

dx/dt +2x = t + 1

t=0,x=1

(i) solve to get x in terms of t (DONE)
(ii)the quantity y is related to x by te D.e 2xdy/dx = y t=0,y=4
solve to find y in terms of x,Henc express y in terms of t

my working: (integral)1/y(dy) = (integral)1/2x(dx)
lny = 0.5ln2x + C
e^lny = e^((2x^0.5)+c)
y = ae^(2x)^0.5

in actual fact a = 4 and this is easily found if separate at start with
(integral)2/y(dy)=(integral)1/x(dx)

this gives 2lny = lnx +c
y^2=e^lnx+c
y^2=Ae^lnx
y=Ax^0.5 initial conditions giving A=4

My method gives basically lny = ln(2x)^0.5 + c
y=Ae^ln(2x)^0.5
y=A(2x)^0.5 initiial conditions giving A=4/root2

If someone could point out my wrong working that would be really helpful ,whether its my initial separation or rule of logs im not sure thnx,

2. Hello,

my working: (integral)1/y(dy) = (integral)1/2x(dx)
lny = 0.5ln2x + C
The problem is that $\int \frac 1{2x} dx=\frac 12 \int \frac 1x dx=\frac 12 \cdot \ln x+c \neq \frac 12 \cdot \ln 2x+c$