Solutions to linear differential equations can be written using convolutions as y = YIvp + (h(t) * f(t)) YIVP is the solution to the associated homogeneous differential equation with the given initial values (ignore the forcing function, keep initial values). h(t) is the impulse response (ignore the initial values and forcing function). • f(t) is the forcing function. (ignore the initial values and differential equation). Use the form above to write the solution to the differential equation y" + 8y + 15y = 6t²e-3t with y(0) = 3, y(0) = -9
Solutions to linear differential equations can be written using convolutions as y = YIvp + (h(t) * f(t)) YIVP is the solution to the associated homogeneous differential equation with the given initial values (ignore the forcing function, keep initial values). h(t) is the impulse response (ignore the initial values and forcing function). • f(t) is the forcing function. (ignore the initial values and differential equation). Use the form above to write the solution to the differential equation y" + 8y + 15y = 6t²e-3t with y(0) = 3, y(0) = -9
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:Solutions to linear differential equations can be written using convolutions as
y = YIVP + (h(t) * f(t))
YIVP is the solution to the associated homogeneous differential equation with the given initial values
(ignore the forcing function, keep initial values).
h(t) is the impulse response
(ignore the initial values and forcing function).
Y
• f(t) is the forcing function.
(ignore the initial values and differential equation).
Use the form above to write the solution to the differential equation
= 13/4
+
3t^2
If you don't get this in 1 tries, you can get a hint.
Hint:
using the characteristic equation
-3t
y" + 8y + 15y = 6t²e-
*6t^(2)e^(-3t)
You can quickly compute the impulse response by converting the impulse at t = 0 to an initial velocity. Solve
y" + 8y + 15y = 0,
with y(0) = 0, y(0) = 1
with y(0) = 3, y(0) = -9
² +8r+15= 0
and plugging in to compute c₁ and c₂. (You could also compute using Laplace transforms, but that is more slow.)
To be accepted, your answer must be entered into webwork as
(Impulse resp. )* (Forcing fun.)
Hint: The forcing function is f(t) = 6t²e-³t
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