Linear Programming

Find the solution of $$\displaystyle 2\left ( x-3y+1 \right )\frac{dy}{dx}= 4x-2y+1.$$
Solve the following Linear Programming Problems graphically :
Minimize $$Z = 3x + 5y$$
subject to
 $$x + y \leq 4, x  \geq 0$$ and $$y \ge 0$$
Solve the differential equation: $$\displaystyle \frac{dy}{dx}-y\tan x= -2\sin x$$
Solve the differential equation: $$\displaystyle \frac{dy}{dx}-y\tan x= -2\sin x$$
If a young man rides his motorcycle at $$25\ km/hr$$, he had to spend $$Rs.2$$ per km on petrol. If he rides at a faster speed of $$40\ km/hr$$, the petrol cost increases at $$Rs.5$$ per km. He has $$Rs.100$$ to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as LPP and solve it graphically.