Second-Order
Linear Homogeneous
Differential Equations with
Constant Coefficients
Part 1: The Equation y"
+ by = 0 with b > 0
- In your worksheet, enter
the starting coefficients (a = 0 and b = 1), the definition
of the differential equation, and the initial conditions, y(0) = 1
and y'(0) = 0. Use the built-in numerical differential equation
solver to sketch a solution of the initial value problem, and study the
solution graph. Increase b to 2, and replot. Now increase
b to 4 and then to 8, each time redrawing the graph.
How do the solutions change as b increases? Use the form of the
symbolic solution to explain the changes you see in the solution
graphs.
- Reset b to 1,
and examine the graphs of the solutions for the initial conditions
- y(0) = 2,
y'(0) = 0, and
- y(0) = -1,
y'(0) = 0.
How do the solutions change
as y(0) varies? Use the symbolic solution to explain what you see
in the solution graphs.
- Reset y(0) = 1,
and change y'(0) first to 2, then to 3. How do the
solutions change as y'(0) varies through positive values? Use the
symbolic solution to explain what you see in the solution graphs.
- Keep y(0) = 1, and
change y'(0) to -2 and then to -3. How do the solutions
change as y'(0) varies through negative values? Use the symbolic
solution to explain what you see in the solution graphs.
