Solve the differential equation Y’(t) = AY(t), with initial
condition Y(0) = [1;0] (a 2x1 matrix); where A = [ 9 , 5 ; -6 , -2
]. Then, using Euler’s method with step size h=.1 over [ 0 , .5 ]
fill in the table with header where the 2x1 matrix Yi is the
approximation of the exact solution Y(ti) :
t Yi Y(ti) ||Y(ti) – Yi ||
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) =...
Consider the differential equation
y′(t)+9y(t)=−4cos(5t)u(t),
with initial condition y(0)=4,
A)Find the Laplace transform of the solution
Y(s).Y(s). Write the solution as a single
fraction in s.
Y(s)= ______________
B) Find the partial fraction decomposition of Y(s). Enter all
factors as first order terms in s, that is, all terms
should be of the form (c/(s-p)), where c is a constant and the root
p is a constant. Both c and p may be complex.
Y(s)= ____ + ______ +______
C)...
Consider the equation: ?̇ +2? = ?(?) with initial condition x(0)
= 2
(a) If u(t) = 0, find the solution ?(?). What is ?(?) as t ->
∞?
(b) If u(t) = 4+t, find the solution ?(?). What is ?(?) as t
-> ∞?
(c) If u(t) = ?3?, find the solution ?(?). What is
?(?) as t -> ∞?
(d) If u(t) = δ(t), find the solution ?(?). What is ?(?) as t
-> ∞?
Find the particular solution of the first-order linear
differential equation for
x > 0
that satisfies the initial condition. (Remember to use
absolute values where appropriate.)
Differential Equation Initial Condition
x dy = (x + y + 7) dx
y(1) = 6
y =
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...
Starting with Maxwell's equations show that the magnetic field
satisfies the same wave equation as the electric field. in
particular, that is, too, propagates with the same speed.