Find the general solution to the homogeneous differential equation The solution has the form with f₁(t) = e- 2t and f₂(t)= e-6t d²y dt² +8 +12y = 0 dt y = C₁fi(t) + C₂f2(t)
Find the general solution to the homogeneous differential equation The solution has the form with f₁(t) = e- 2t and f₂(t)= e-6t d²y dt² +8 +12y = 0 dt y = C₁fi(t) + C₂f2(t)
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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![**Finding the General Solution to a Homogeneous Differential Equation**
To find the general solution for the homogeneous differential equation:
\[
\frac{d^2y}{dt^2} + 8\frac{dy}{dt} + 12y = 0
\]
We express the solution in the form:
\[
y = C_1 f_1(t) + C_2 f_2(t)
\]
where:
\[
f_1(t) = e^{-2t}
\]
and
\[
f_2(t) = e^{-6t}
\]
**Explanation:**
The differential equation given involves a second-order linear homogeneous equation with constant coefficients. The solution involves finding the roots of the characteristic equation, which leads to the exponential solutions \( e^{-2t} \) and \( e^{-6t} \). The general solution is a linear combination of these functions with constants \( C_1 \) and \( C_2 \) that can be determined by initial conditions if provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccc4b4bb-a94d-4dc4-be7d-92733a14ab19%2F67e19a3b-46a4-42bb-a2b2-3b026b67ed3c%2Fetzdxrc_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the General Solution to a Homogeneous Differential Equation**
To find the general solution for the homogeneous differential equation:
\[
\frac{d^2y}{dt^2} + 8\frac{dy}{dt} + 12y = 0
\]
We express the solution in the form:
\[
y = C_1 f_1(t) + C_2 f_2(t)
\]
where:
\[
f_1(t) = e^{-2t}
\]
and
\[
f_2(t) = e^{-6t}
\]
**Explanation:**
The differential equation given involves a second-order linear homogeneous equation with constant coefficients. The solution involves finding the roots of the characteristic equation, which leads to the exponential solutions \( e^{-2t} \) and \( e^{-6t} \). The general solution is a linear combination of these functions with constants \( C_1 \) and \( C_2 \) that can be determined by initial conditions if provided.
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this is wrong
f1 does not equal e- 2t and f2 does not equal e - 6t
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