1. ## quotient rule differenciate

b(i) use the Quotient Rule to differentiate the function

h(x) = 1+Inx
x (x>0).

ii) using your answer to part (b) (i), find the general solution of the differential equation

dy/dx=(-inx/x^2)(y^1/2) (x>0,y>0)

Give the solution in implicit form.

(iii) Find the particular solution of the differential equation in part (b) (ii) for which y=1 when x=1, and then give this particular solution in explicit form.

2. Originally Posted by student01.06

b(i) use the Quotient Rule to differentiate the function

h(x) = 1+Inx
x (x>0).
Unfortunately I am not able to assist you with the differential equation part, but I can help you with the quotient rule.

The quotient rule is the following:

$\displaystyle \left( \frac{f}{g} \right) ^{'} (x) = \frac{f'(x)g(x) - f(x)g'(x)}{\left[ g(x) \right] ^2}$

$\displaystyle h(x) = \frac{1+ \ln x}{x}$

$\displaystyle h'(x) = \frac{(1+ \ln x)'(x) - (1 + \ln x)(x)'}{x^2} = \frac{(\frac{1}{x})(x) - (1 + \ln x)(1)}{x^2} = \frac{- \ln x}{x^2}$

3. Originally Posted by student01.06
ii) using your answer to part (b) (i), find the general solution of the differential equation

dy/dx=(-inx/x^2)(y^1/2) (x>0,y>0)

Give the solution in implicit form.
Separation of variables:

$\displaystyle \frac{dy}{dx} = \frac{\left(-\ln x\right)\sqrt y}{x^2}$

$\displaystyle \Rightarrow\int\frac{dy}{\sqrt y} = \int\left(-\frac{\ln x}{x^2}\right)dx$

Now, do you notice something about the integrand on the right?