find and plot both the steady periodic solution
x 
sp
​
 (t)=Ccos(ωt−α)
of the given differential equation and the actual solution
x(t)=x 
sp
​
 (t)+x 
tr
​
 (t)
that satisfies the given initial conditions. 11.
x 
′′
 +4x 
′
 +5x=10cos3t;x(0)=x 
′
 (0)=0

Here's the plot of the solutions:

- The **dashed line** represents the **steady periodic solution** \( x_{sp}(t) = \frac{25}{9} \cos(3t) - \frac{5}{9} \sin(3t) \).
- The **solid blue line** represents the **actual solution** \( x(t) = x_{sp}(t) + x_{tr}(t) \).

The actual solution \( x(t) \) combines the steady periodic component and the transient component, which decays over time due to the exponential factor \( e^{-2t} \).

Would you like more details or have any questions?

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Here are 5 related questions to explore further:

1. How would the solution change if the initial conditions were different?
2. What role does the damping term (coefficient of \( x' \)) play in the solution?
3. How would the solution behave if the driving frequency was different from 3?
4. Can you derive the solution for a different forcing function, such as \( \sin(3t) \)?
5. What happens to the solution as \( t \) approaches infinity?

**Tip:** Understanding the steady-state and transient components of a solution helps in analyzing how systems respond to external forces over time.

Explore the plot of steady periodic solution x_sp(t) = C*cos(ωt - α) and actual solution x(t) = x_sp(t) + x_tr(t) for the differential equation x'' + 4x' + 5x = 10cos(3t). Understand how the solutions evolve over time with given initial conditions x(0) = 0 and x'(0) = 0.

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