4. Find the solution of the IVP y" + 4y = g(t), y(0)=0, y′(0)= 0, where 0≤t <10 (0, g(t)={(t−5)/5, 5≤t<10. t≥10
4. Find the solution of the IVP y" + 4y = g(t), y(0)=0, y′(0)= 0, where 0≤t <10 (0, g(t)={(t−5)/5, 5≤t<10. t≥10
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 54CR
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![**Problem Statement:**
Find the solution of the Initial Value Problem (IVP):
\[ y'' + 4y = g(t), \]
\[ y(0) = 0, \]
\[ y'(0) = 0, \]
where
\[ g(t) = \begin{cases}
0, & 0 \le t < 10 \\
(t - 5)/5, & 5 \le t < 10 \\
1, & t \ge 10
\end{cases} \]
**Explanation:**
This is an Ordinary Differential Equation (ODE) with given initial conditions. The function \( g(t) \) is piecewise-defined with different expressions for different intervals of the variable \( t \):
1. \( g(t) = 0 \) for \( 0 \le t < 10 \).
2. \( g(t) = (t - 5)/5 \) for \( 5 \le t < 10 \).
3. \( g(t) = 1 \) for \( t \ge 10 \).
To solve this problem, you need to integrate the piecewise-defined function \( g(t) \) and apply the initial conditions to find a specific solution \( y(t) \). The solution process involves dividing the problem into intervals and solving for each interval separately, then ensuring continuity at the boundaries (\( t = 5 \) and \( t = 10 \)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55171999-1bf3-4f89-b6f4-3fd68bbe2f25%2Fd097084a-c78b-4fa2-a4a4-073de947b235%2Ff37wwk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the solution of the Initial Value Problem (IVP):
\[ y'' + 4y = g(t), \]
\[ y(0) = 0, \]
\[ y'(0) = 0, \]
where
\[ g(t) = \begin{cases}
0, & 0 \le t < 10 \\
(t - 5)/5, & 5 \le t < 10 \\
1, & t \ge 10
\end{cases} \]
**Explanation:**
This is an Ordinary Differential Equation (ODE) with given initial conditions. The function \( g(t) \) is piecewise-defined with different expressions for different intervals of the variable \( t \):
1. \( g(t) = 0 \) for \( 0 \le t < 10 \).
2. \( g(t) = (t - 5)/5 \) for \( 5 \le t < 10 \).
3. \( g(t) = 1 \) for \( t \ge 10 \).
To solve this problem, you need to integrate the piecewise-defined function \( g(t) \) and apply the initial conditions to find a specific solution \( y(t) \). The solution process involves dividing the problem into intervals and solving for each interval separately, then ensuring continuity at the boundaries (\( t = 5 \) and \( t = 10 \)).
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