Determine the full solution of the ordinary differential equation (ODE) given below. The usual form of the particular solution is provided as help. *(t) + 2x(t) + 4x(t) = cos(2t), Initial conditions: x(0) = 0, x(0) = 0 Usual Form of the Particular Solution TABLE F(t) a ait + ao €at Coswt sin wt yp (1) A At + B Aeat A coswt + B sinwt Acoswt +B sin wt
Determine the full solution of the ordinary differential equation (ODE) given below. The usual form of the particular solution is provided as help. *(t) + 2x(t) + 4x(t) = cos(2t), Initial conditions: x(0) = 0, x(0) = 0 Usual Form of the Particular Solution TABLE F(t) a ait + ao €at Coswt sin wt yp (1) A At + B Aeat A coswt + B sinwt Acoswt +B sin wt
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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![**Problem Statement:**
Determine the full solution of the ordinary differential equation (ODE) given below. The usual form of the particular solution is provided as help.
\[
\ddot{x}(t) + 2\dot{x}(t) + 4x(t) = \cos(2t)
\]
**Initial conditions:**
\[ x(0) = 0, \quad \dot{x}(0) = 0 \]
**Table: Usual Form of the Particular Solution**
| \( F(t) \) | \( y_p(t) \) |
|-----------------------|-------------------------------|
| \( \alpha \) | \( A \) |
| \( \alpha_1 t + \alpha_0 \) | \( At + B \) |
| \( e^{\alpha t} \) | \( Ae^{\alpha t} \) |
| \( \cos \omega t \) | \( A \cos \omega t + B \sin \omega t \) |
| \( \sin \omega t \) | \( A \cos \omega t + B \sin \omega t \) |
This table provides forms for guessing particular solutions in cases where the non-homogeneous term \( F(t) \) takes on specific functional forms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53a2441e-c8b0-4860-bf8a-5f8d74e1cf79%2F80452e1f-9e0d-4f70-b53a-3aafc2e166f1%2Fehydpkr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine the full solution of the ordinary differential equation (ODE) given below. The usual form of the particular solution is provided as help.
\[
\ddot{x}(t) + 2\dot{x}(t) + 4x(t) = \cos(2t)
\]
**Initial conditions:**
\[ x(0) = 0, \quad \dot{x}(0) = 0 \]
**Table: Usual Form of the Particular Solution**
| \( F(t) \) | \( y_p(t) \) |
|-----------------------|-------------------------------|
| \( \alpha \) | \( A \) |
| \( \alpha_1 t + \alpha_0 \) | \( At + B \) |
| \( e^{\alpha t} \) | \( Ae^{\alpha t} \) |
| \( \cos \omega t \) | \( A \cos \omega t + B \sin \omega t \) |
| \( \sin \omega t \) | \( A \cos \omega t + B \sin \omega t \) |
This table provides forms for guessing particular solutions in cases where the non-homogeneous term \( F(t) \) takes on specific functional forms.
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