Problem 4 (Non-homogeneous Equations Redux). By now we are very procedure for solving a linear, second-order initial value problem y" + p(t)y' + q(t)y = f(t), y(to) = y0, y'(to) = vo- %3D • First find the general solution to the associated homogeneous equati- y" + p(t)y/ + q(t)y = 0. %3D • Find a particular solution to the non-homogeneous equation y" + p(t)y' + q(t)y = f(t) %3D and add it to the homogeneous solution. • Plug in the initial conditions to find the solution to the IVP. There is actually another way we can proceed, and it allows us to plug the into the homogeneous solution before finding the particular solution. (a) Suppose u1(t) is a solution to the initial value problem y" + p(t)y' + q(t)y = 0, y(to) = yo, y (to) = %3D %3D and u2(t) is a solution to y" + p(t)y' + q(t)y = f(t), y(to) = 0, y (to) = 0. %3D %3D %3D Show that y(t) = u1 (t) + u2 (t) is a solution to the original initial val %3D
Problem 4 (Non-homogeneous Equations Redux). By now we are very procedure for solving a linear, second-order initial value problem y" + p(t)y' + q(t)y = f(t), y(to) = y0, y'(to) = vo- %3D • First find the general solution to the associated homogeneous equati- y" + p(t)y/ + q(t)y = 0. %3D • Find a particular solution to the non-homogeneous equation y" + p(t)y' + q(t)y = f(t) %3D and add it to the homogeneous solution. • Plug in the initial conditions to find the solution to the IVP. There is actually another way we can proceed, and it allows us to plug the into the homogeneous solution before finding the particular solution. (a) Suppose u1(t) is a solution to the initial value problem y" + p(t)y' + q(t)y = 0, y(to) = yo, y (to) = %3D %3D and u2(t) is a solution to y" + p(t)y' + q(t)y = f(t), y(to) = 0, y (to) = 0. %3D %3D %3D Show that y(t) = u1 (t) + u2 (t) is a solution to the original initial val %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 4 (Non-homogeneous Equations Redux). By now we are very
procedure for solving a linear, second-order initial value problem
y" + p(t)y' + q(t)y = f(t), y(to) = y0, y'(to) = vo-
%3D
• First find the general solution to the associated homogeneous equati-
y" + p(t)y/ + q(t)y = 0.
%3D
• Find a particular solution to the non-homogeneous equation
y" + p(t)y' + q(t)y = f(t)
%3D
and add it to the homogeneous solution.
• Plug in the initial conditions to find the solution to the IVP.
There is actually another way we can proceed, and it allows us to plug the
into the homogeneous solution before finding the particular solution.
(a) Suppose u1(t) is a solution to the initial value problem
y" + p(t)y' + q(t)y = 0, y(to) = yo, y (to) =
%3D
%3D
and u2(t) is a solution to
y" + p(t)y' + q(t)y = f(t), y(to) = 0, y (to) = 0.
%3D
%3D
%3D
Show that y(t) = u1 (t) + u2 (t) is a solution to the original initial val
%3D
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