M427K Differential Equations Exam 1 – Sept 21,2012
1. Solve the differential equation
xy 0 + 2y = 4x2
in 3 different cases: a)y(0) = 0; b) y(1) = 0; c) y(1) = 2
Make a crude sketch of the graph in each case.
How might you characterize the solutions to this differential equation?
2. Solve the differential equation
(y cos(x) + 2xey ) + (sin(x) + x2 ey − 1)y 0 = 0
with y(π/2 = 1.
3. Solve the differential equation
y” + 5y 0 + 6y = 0
with y(0) = 2, y 0 (0) = 3.
How does the solution behave as x− > ∞ ?
Using the initial data y(0) = 2, y 0 (0) = 3 make a very crude sketch of the solution curve
for 0 < x < ∞.
4. Attached is the direction field plot for a 1st order differential eauation with several
solutions marked. What conclusions can you draw about the differential equation from
looking at this plot?