3/" + p(t)y' + q(t)y = f(t), y(to) = y0, y'(to) = vo- (1) %3D %3D • First find the general solution to the associated homogeneous equation 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 initial conditions 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) = vo, %3D and u2(t) is a solution to y" + p(t)y' + q(t)y = f(t), y(to) = 0, y(to) = 0. (2) %3D Show that y(t) = u1 (t) + u2 (t) is a solution to the original initial value problem (1). %3D
3/" + p(t)y' + q(t)y = f(t), y(to) = y0, y'(to) = vo- (1) %3D %3D • First find the general solution to the associated homogeneous equation 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 initial conditions 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) = vo, %3D and u2(t) is a solution to y" + p(t)y' + q(t)y = f(t), y(to) = 0, y(to) = 0. (2) %3D Show that y(t) = u1 (t) + u2 (t) is a solution to the original initial value problem (1). %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:3/" + p(t)y' + q(t)y = f(t), y(to) = y0, y'(to) = vo-
(1)
%3D
%3D
• First find the general solution to the associated homogeneous equation
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 initial conditions
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) = vo,
%3D
and u2(t) is a solution to
y" + p(t)y' + q(t)y = f(t), y(to) = 0, y(to) = 0.
(2)
%3D
Show that y(t) = u1 (t) + u2 (t) is a solution to the original initial value problem (1).
%3D
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