# differentiation, separation of variables???

• May 11th 2008, 07:57 AM
i_zz_y_ill
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,
• May 11th 2008, 08:42 AM
Moo
Hello,

Quote:

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