The First Quiz

Students of the Ordinary Differential Equations course had their first quiz on Monday, August 29, 2011 - exactly one week into their first semester. There was some consternation till it was pointed out that an early quiz is necessarily an easy quiz.

Here are the questions and their solutions:

1. Give the order and degree of $ 2x^2y^{\prime\prime} -3y^\prime +y=0$.
   
   Solution: The highest order derivative present is of second order ($y^{\prime\prime}$), so the ODE has order 2. The highest order derivative is present with degree 1, so the ODE has degree 1.
   
   
2. 
   
   
   1. Verify that $y(x)= ce^{-x}+2$ is a general solution to $y^\prime +y =2$.
      
      Solution: We calculate $y^\prime = -ce^{-x}$. Now we substitute $y$ and $y^\prime$ into the ODE:
      \[ LHS = y^\prime +y = -ce^{-x} + ce^{-x}+2 = 2 = RHS\]
      So the ODE is satisfied and the given $y$ is a general solution.
   2. Find the particular solution to this ODE given that $y=3.2$ when $x=0$.
      
      Solution: Substitute this pair of values into the general solution:
      \[ce^{-0} + 2 = 3.2\]
      We solve this for $c$ and get $c=1.2$. So the particular solution for the given initial condition is $y= 1.2 e^{-x}+2$.
      
   3. Graph this solution.
      
      Solution: