Homework Answers
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a)...
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a) Check that y(t) = 1-1 is a solution to this equation. (b) Find another solution (t) such that yı(t) and (t) are linearly independent (that is, wit) and y(t) form a fundamental set of solutions for the differential equation).
Consider the following IVP y″ + 5y′ + y = f (t), y(0) = 3, y′(0) ...
Consider the following IVP
y″ + 5y′ +
y = f (t), y(0) = 3,
y′(0) = 0,
where
f (t) =
{
8
0 ≤ t ≤ 2π
cos(7t)
t > 2π
(a)
Find the Laplace transform F(s) =
ℒ { f (t)} of f (t).
(b)
Find the Laplace transform Y(s) =
ℒ {y(t)} of the solution y(t)
of the above IVP.
Consider the following IVP y" + 5y' + y = f(t), y(0) = 3, y'(0) =...
Consider the following initial value problem у(0) — 0. у%3D х+ у, (i) Solve the differential equation above in tabul...
Consider the following initial value problem у(0) — 0. у%3D х+ у, (i) Solve the differential equation above in tabular form with h= 0.2 to approximate the solution at x=1 by using Euler's method. Give your answer accurate to 4 decimal places. Given the exact solution of the differential equation above is y= e-x-1. Calculate (ii) all the error and percentage of relative error between the exact and the approximate y values for each of values in (i) 0.2 0.4...
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use...
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use Abel's formula to find the Wronskian of any two solutions of this equation and W[y1,y2](t). What do you observe? compare it to = t1 and y2(t) = t-1 nt represent a fundamental set of solu f) (2 points) Determine if y1 (t) tions (2 points) Find the general solution of t2y" +3ty' +y = 0. g) Solve the initial value problem t2y" + 3ty/...
(1 point) Consider the initial value problem y" + 4y = 8t, y(0) =3, y'(0) =...
(1 point) Consider the initial value problem y" + 4y = 8t, y(0) =3, y'(0) = 4. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). 8/s^2 help (formulas) b. Solve your equation for Y(s). Y(s) = L{y(t)} = c. Take...
Question 1: [25 pts] Consider the IVP y" – 4y' - 5y = 0, y(0) =...
Question 1: [25 pts] Consider the IVP y" – 4y' - 5y = 0, y(0) = 1, y0) = 2. a) Find the solution of the given IVP using the corresponding characteristic equation. b) Find the solution of the IVP using the Laplace Transform. c) Does the solution change if we would change the second initial condition as y'(0)=3? Explain.
(1 point) Consider the initial value problem y' + 3y = 0 if 0 <t <3...
(1 point) Consider the initial value problem y' + 3y = 0 if 0 <t <3 9 if 3 < t < 5 0 if 5 <t< oo, y(0) = 3. (a) Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y by Y. Do not move any terms from one side of the equation to the other (until you get to part (b) below). y(s)(5+6)...
(1 point) Consider the initial value problem 4y 8t, y(0) 4, y'(0) 3. f both sides of the given differential equatio...
(1 point) Consider the initial value problem 4y 8t, y(0) 4, y'(0) 3. f both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from othe other (until you get to part (b) below). Take the Laplace transform one side of the equation help (formulas) b. Solve your equation for Y(8) Y(s) C{y(t) = Take the inverse Laplace transform of both sides of the...
(1 point) Take the Laplace transform of the IVP dy -1 di - y= 0, y(0)...
(1 point) Take the Laplace transform of the IVP dy -1 di - y= 0, y(0) = 6 Use Y for the Laplace transform of y, (not Y(s)). II So Y= S- and y(t) =
Consider the following initial value problem. y′ + 5y = { 0 t ≤ 1 10...
Consider the following initial value problem.
y′ + 5y =
{
0
t ≤ 1
10
1 ≤ t < 6
0
6 ≤ t < ∞
y(0) = 4
(a)
Find the Laplace transform of the right hand side of the above
differential equation.
(b)
Let y(t) denote the solution to the above
differential equation, and let Y((s) denote the
Laplace transform of y(t). Find
Y(s).
(c)
By taking the inverse Laplace transform of your answer to (b),
the...
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a) Check that y(t) = 1-1 is a solution to this equation. (b) Find another solution (t) such that yı(t) and (t) are linearly independent (that is, wit) and y(t) form a fundamental set of solutions for the differential equation).
Consider the following IVP
y″ + 5y′ +
y = f (t), y(0) = 3,
y′(0) = 0,
where
f (t) =
{
8
0 ≤ t ≤ 2π
cos(7t)
t > 2π
(a)
Find the Laplace transform F(s) =
ℒ { f (t)} of f (t).
(b)
Find the Laplace transform Y(s) =
ℒ {y(t)} of the solution y(t)
of the above IVP.
Consider the following IVP y" + 5y' + y = f(t), y(0) = 3, y'(0) =...
Consider the following initial value problem у(0) — 0. у%3D х+ у, (i) Solve the differential equation above in tabular form with h= 0.2 to approximate the solution at x=1 by using Euler's method. Give your answer accurate to 4 decimal places. Given the exact solution of the differential equation above is y= e-x-1. Calculate (ii) all the error and percentage of relative error between the exact and the approximate y values for each of values in (i) 0.2 0.4...
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use Abel's formula to find the Wronskian of any two solutions of this equation and W[y1,y2](t). What do you observe? compare it to = t1 and y2(t) = t-1 nt represent a fundamental set of solu f) (2 points) Determine if y1 (t) tions (2 points) Find the general solution of t2y" +3ty' +y = 0. g) Solve the initial value problem t2y" + 3ty/...
(1 point) Consider the initial value problem y" + 4y = 8t, y(0) =3, y'(0) = 4. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). 8/s^2 help (formulas) b. Solve your equation for Y(s). Y(s) = L{y(t)} = c. Take...
Question 1: [25 pts] Consider the IVP y" – 4y' - 5y = 0, y(0) = 1, y0) = 2. a) Find the solution of the given IVP using the corresponding characteristic equation. b) Find the solution of the IVP using the Laplace Transform. c) Does the solution change if we would change the second initial condition as y'(0)=3? Explain.
(1 point) Consider the initial value problem y' + 3y = 0 if 0 <t <3 9 if 3 < t < 5 0 if 5 <t< oo, y(0) = 3. (a) Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y by Y. Do not move any terms from one side of the equation to the other (until you get to part (b) below). y(s)(5+6)...
(1 point) Consider the initial value problem 4y 8t, y(0) 4, y'(0) 3. f both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from othe other (until you get to part (b) below). Take the Laplace transform one side of the equation help (formulas) b. Solve your equation for Y(8) Y(s) C{y(t) = Take the inverse Laplace transform of both sides of the...
(1 point) Take the Laplace transform of the IVP dy -1 di - y= 0, y(0) = 6 Use Y for the Laplace transform of y, (not Y(s)). II So Y= S- and y(t) =
Consider the following initial value problem.
y′ + 5y =
{
0
t ≤ 1
10
1 ≤ t < 6
0
6 ≤ t < ∞
y(0) = 4
(a)
Find the Laplace transform of the right hand side of the above
differential equation.
(b)
Let y(t) denote the solution to the above
differential equation, and let Y((s) denote the
Laplace transform of y(t). Find
Y(s).
(c)
By taking the inverse Laplace transform of your answer to (b),
the...
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