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There are two questions that I am stuck on.
The first one is this:
By using the substitution y=xu, find the equation of the curve which passes through (1,0) and satisfies the differential equation dy/dx=2x/x+y.
Here is my working
dy/dx=xdu/dx+u
(sub into the equation)
xdu/dx+u=2x/x(1+u)
xdu/dx=2/(1+u) - u
xdu/dx=2-u-u²/(1+u)
(splitting the variables)
-(1+u)/u²+u-2 du = 1/x dx
On the RHS, I split it into partial fractions to get:
-1/3(1/u+2 + 2/u-1)du = lnx + c
-1/3(ln|u+2| + ln|u-1|²) = lnx + c
ln(u+2)(u-1)² = -3lnx + c'
(u+2)(u-1)² = 1/x³ + c''
Subbing u=y/x into the whole equation:
(y/x + 2)(y/x - 1)² = 1/x³ + c
After lots of rearranging, I get
Oh wait.... never mind. I got it now. XD
Oh dear........
Well, there is this second question which says:
A car move sfrom rest along a straigh road. After t seconds the velocity is v metres per second. The motion is modelled by dv/dt + av = e^(bt), where a and p are positive constants.
Find v in terms of a, b and t and show that, as long as the aboe model applies, the car does not come to rest.
Please could you point in the right direction of this question, please?
Thank you very much, in advance.
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