6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1
.
(a) Use Euler’s method (by hand) to approximate the solution
y(t) at t = 2 using ∆t = 1, ∆t = 1/2 and ∆t = 1/4 (use a calculator
to approximate the answer for the smallest ∆t). Report your results
in a table listing ∆t in one column, and the corresponding
approximation of y(2) in the other.
(b) The direction field of dy/dt...
3. Consider the IVP:
dy =ty^1/3; y(0)=0,t≥0. dt
Both y(t) = 0, (the equilibrium solution) and y(t) =
?(1/3t^2?)^3/2 are solutions to this IVP.
(a) Show that the trivial solution satisfies the IVP by first
verifying that it satisfies the initial condition and then
verifying that it satisfies the differential equation.
(b) Show that the other solution satisfies the IVP again by
first verifying it satisfies the initial condition and then
verifying that it satisfies the differential equation.
(c) Explain...
Consider the system modeled by the differential equation
dy/dt - y = t with initial condition y(0) = 1
the exact solution is given by y(t) = 2et − t − 1
Note, the differential equation dy/dt - y =t can be written as
dy/dt = t + y
using Euler’s approximation of dy/dt = (y(t + Dt) – y(t))/ Dt
(y(t + Dt) – y(t))/ Dt = (t + y)
y(t + Dt) =...
Initial value problem : Differential equations:
dx/dt = x + 2y
dy/dt = 2x + y
Initial conditions:
x(0) = 0
y(0) = 2
a) Find the solution to this initial value problem
(yes, I know, the text says that the solutions are
x(t)= e^3t - e^-t and y(x) = e^3t + e^-t
and but I want you to derive these solutions yourself using one
of the methods we studied in chapter 4) Work this part out on paper
to...
Consider the differential equation dy/dt = 2?square
root(absolute value of y) with initial condition y(t0)=y0
• For what values of y0 does the Existence Theorem apply?
• For what values of y0 does the Uniqueness theorem apply?
• Verify that y1(t) = 0 solves the initial value problem with y0 =
0
• Verify that y2(t) = t2 solves the initial value problem with y0 =
0
• Does this violate the theorems from this section 1.5? Why or why...