### Linear Programming

goals
A dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin A, $12$ units of vitamin B and $8$ units of vitamin C. The vitamin contents of one kg food is given below:
 Food Vitamin A Vitamin B Vitamin C $X$ $1$ $2$ $3$ $Y$ $2$ $2$ $1$
One kg of food $X$ costs $Rs. 16$ and one kg of food $Y$ costs $Rs. 20$. Find the least cost of the mixture which will produce the required diet?
Find the general solution of the differential equation:
$xdy-ydx=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } dx$

Find the general solution of the differential equation:
$xdy-ydx=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } dx$

Holiday Mean Turkey Ranch is considering buying two different types of turkey feed. Each feed contains, in varying proportions, some or all of the three nutritional ingredients essential for fattening turkeys. Brand Y feed costs the ranch $0.2$ per pound. Brand Z costs $.03$ per pound. The rancher would like to determine the lowest-cost diet that meets the minimum monthly intake requirement for each nutritional ingredient.
The following table contains relevant information about the composition of brand Y and brand Z feeds, as well as the minimum monthly requirement for each nutritional ingredient per turkey.
Composition of Each Pound of Feed
 Ingredient Brand Y Feed Brand Z Feed Minimum Monthly Requirement A $5$ oz $10$ oz $90$ oz B $4$ oz $3$ oz $48$ oz C $.5$ oz $0$ $1.5$ oz Cost/lb $0.2$ $.03$
Solve : $\dfrac{dy}{dx} = 1 + x + y + xy$