Find the general solution of this ODE: d²y dy +2- +y=12e-t dt² dt The solution will be of the form: y(t) = Cy₁(t) + Dy₂(t) + yp(t)
Find the general solution of this ODE: d²y dy +2- +y=12e-t dt² dt The solution will be of the form: y(t) = Cy₁(t) + Dy₂(t) + yp(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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Please explain each step in detail, so I may understand properly.
![### General Solution of a Second-Order Ordinary Differential Equation (ODE)
**Problem Statement:**
Find the general solution of this ODE:
\[
\frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + y = 12e^{-t}
\]
**Solution Approach:**
The solution will be of the form:
\[
y(t) = C y_1(t) + D y_2(t) + y_p(t)
\]
where \( C \) and \( D \) are arbitrary constants.
**Instructions:**
To solve this equation, follow these steps:
1. Find the complementary solution \( y_c(t) \) by solving the homogeneous equation.
2. Determine particular solution \( y_p(t) \) using an appropriate method such as undetermined coefficients or variation of parameters.
3. Combine these results to express the general solution.
The final general solution in the form:
\[
y(t) =
\]
In this context, the arbitrary constants \( C \) and \( D \) will adjust based on initial or boundary conditions provided in a specific problem scenario.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2F944cda40-835d-43cd-83d0-1ebf3c091095%2F0oc7udk_processed.png&w=3840&q=75)
Transcribed Image Text:### General Solution of a Second-Order Ordinary Differential Equation (ODE)
**Problem Statement:**
Find the general solution of this ODE:
\[
\frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + y = 12e^{-t}
\]
**Solution Approach:**
The solution will be of the form:
\[
y(t) = C y_1(t) + D y_2(t) + y_p(t)
\]
where \( C \) and \( D \) are arbitrary constants.
**Instructions:**
To solve this equation, follow these steps:
1. Find the complementary solution \( y_c(t) \) by solving the homogeneous equation.
2. Determine particular solution \( y_p(t) \) using an appropriate method such as undetermined coefficients or variation of parameters.
3. Combine these results to express the general solution.
The final general solution in the form:
\[
y(t) =
\]
In this context, the arbitrary constants \( C \) and \( D \) will adjust based on initial or boundary conditions provided in a specific problem scenario.
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the answer that was given was y(t)=Ce^-t + Dte^-t +3t^2e^-t but its wrong I'm looking for the error but can find it
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