Exam 4

1. Match the slope fields with their differential equations. (20%)
   1. $y^{\prime} = y^{2}$
   2. $y^{\prime} = x-y$
   3. $y^{\prime} = \cos x$
   4. $y^{\prime} = x e^{-x}$
   
   

2. Find the solutions to the following differential equations. (15% each)

   1. $$\frac{d y}{d x} = 0.5 (y - 200), \quad y(0) = 50 .$$

   2. $$\frac{d y}{d x} = \frac{5y}{x}, \quad y(1) = 3 .$$

   3. $$t \frac{dx}{dt} = (1 + 2 \ln t) \tan x .$$

   4. $$y^{\prime\prime} - 3 y^{\prime} - 4 y = 0, \quad y(0)=1, \ y^{\prime}(0) = 0 .$$

   5. $$y^{\prime\prime} + 4 y^{\prime} + 5 y = 0, \quad y(0)=1, \ y^{\prime}(\pi/2) = 5 .$$

3. Which differential equation(s) cannot have a solution as shown in the graph? (5%)
   
   1. $y^{\prime\prime} + 2 y^{\prime} +2 y = 0$
   2. $y^{\prime\prime} = 9y$
   3. $y^{\prime\prime} + 4 y^{\prime} +13 y = 0$
   4. $y^{\prime\prime} + 3 y^{\prime} +2 y = 0$