ODE - Assignment 1

Due Date: September 9 September 12

1. A tank contains 800 gal of water in which 200 lb of salt is dissolved. Two gallons of fresh water runs in per minute and 2 gal of the mixture in the tank, kept uniform by stirring, runs out per minute.
   
   1. Form a differential equation for the amount of salt in the tank as a function of time.
   2. How much salt is left in the tank after 5 hours?
2. A thermometer, reading $10^\circ\mathrm{C}$, is brought into a room whose temperature is $23^\circ\mathrm{C}$. Two minutes later the thermometer reading is $18^\circ\mathrm{C}$.
   
   1. Use Newton's Law of Cooling to model the change of the thermometer reading with time.
   2. How long will it take until the reading is practically $23^\circ\mathrm{C}$, say, $22.8^\circ\mathrm{C}$?
3. Solve the following initial value problem (IVP) and make a reasonably accurate sketch of the solution:
   \[ 2y^\prime + y^3 = 0;\quad y(0)=1 \]
   
4. Solve the following ODEs:
   
   1. $\left[\sin(y)\cos(y)+x\cos^2(y)\right]\,dx + x\,dy=0$
   2. $e^y\,\left[\sinh(x)\,dx + \cosh(x)\,dy\right]=0$
   3. $(x^2+y^2)\,dx - 2xy\,dy =0$
   4. $\left(\cos(xy)+\dfrac{x}{y}\right)\,dx + \left(1 + \dfrac{x}{y}\cos(xy)\right)\,dy=0$
5. 1. Under what conditions for the constants $A$,$B$,$C$,$D$, is the following ODE exact?
      \[(Ax+By)\,dx + (Cx+Dy)\,dy=0\]
   2. Solve this exact ODE.
6. 1. Solve the ordinary differential equation: $y^\prime\tan(x)=2y-8$.
   2. Sketch the family of solutions given by the general solution to the above ODE.
   3. Give the particular solution to the above ODE such that $y=0$ when $x=\pi/2$.