Please solve & show steps... **Initial Value Problem Solution**Given the initial value problem:\[ y' - y = 9 + 6\sin t, \quad y(0) = y_0 \]We need to find the value of $ y_0 $ for which the solution $ y(t) $ remains finite as $ t \to \infty $.**Solution Method:**1. **Identify the differential equation:** The given differential equation is $ y' - y = 9 + 6\sin t $.2. **Find the homogeneous solution:** Solve the homogeneous part of the equation $ y' - y = 0 $ to get the general solution $ y_h(t) $.3. **Find the particular solution:** Solve for a particular solution $ y_p(t) $ to the non-homogeneous equation.4. **Combine solutions:** The general solution to the differential equation will be the sum of the homogeneous and particular solutions.5. **Apply initial condition:** Use the initial condition $ y(0) = y_0 $ to determine the specific value of the constant in the general solution.6. **Ensure finiteness as $ t \to \infty $:** Adjust the value of $ y_0 $ as needed to ensure the solution remains finite as $ t \to \infty $.**Solution Box:**\[ y_0 = \boxed{\phantom{some_value}} \]Fill in the box with the appropriate value of $ y_0 $ you find from your calculations to ensure the solution $ y(t) $ remains finite as $ t \to \infty $.

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Description: Please solve & show steps... **Initial Value Problem Solution**Given the initial value problem:\[ y' - y = 9 + 6\sin t, \quad y(0) = y_0 \]We need to find the value of $ y_0 $ for which the solution $ y(t) $ remains finite as $ t \to \infty $.*

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Answered: Find the value of yo for which the…

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Find the value of yo for which the solution of the initial value problem y' – y = 9+ 6 sin t, y(0) = yo remains finite as t →0. %3D Yo ||

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 solve & show steps...

**Initial Value Problem Solution**

Given the initial value problem:
\[ y' - y = 9 + 6\sin t, \quad y(0) = y_0 \]

We need to find the value of $ y_0 $ for which the solution $ y(t) $ remains finite as $ t \to \infty $.

**Solution Method:**

1. **Identify the differential equation:**
The given differential equation is $ y' - y = 9 + 6\sin t $.

2. **Find the homogeneous solution:**
Solve the homogeneous part of the equation $ y' - y = 0 $ to get the general solution $ y_h(t) $.

3. **Find the particular solution:**
Solve for a particular solution $ y_p(t) $ to the non-homogeneous equation.

4. **Combine solutions:**
The general solution to the differential equation will be the sum of the homogeneous and particular solutions.

5. **Apply initial condition:**
Use the initial condition $ y(0) = y_0 $ to determine the specific value of the constant in the general solution.

6. **Ensure finiteness as $ t \to \infty $:**
Adjust the value of $ y_0 $ as needed to ensure the solution remains finite as $ t \to \infty $.

**Solution Box:**

\[ y_0 = \boxed{\phantom{some_value}} \]

Fill in the box with the appropriate value of $ y_0 $ you find from your calculations to ensure the solution $ y(t) $ remains finite as $ t \to \infty $.

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