Find Laplace transform of the function: : f(t)=yt^3 +4

To find the Laplace transform of the function \( f(t) = y t^3 + 4 \), we use the following properties and standard transforms:

1. **Linearity of the Laplace Transform:**
   \[
   \mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\}
   \]
   where \( a \) and \( b \) are constants.

2. **Laplace Transform of Basic Functions:**
   - For \( t^n \) where \( n \) is a positive integer:
     \[
     \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}
     \]
   - For a constant \( c \):
     \[
     \mathcal{L}\{c\} = \frac{c}{s}
     \]

Now, applying these to the given function:

1. **Laplace Transform of \( y t^3 \):**
   \[
   \mathcal{L}\{y t^3\} = y \cdot \mathcal{L}\{t^3\} = y \cdot \frac{3!}{s^{4}} = \frac{6y}{s^4}
   \]

2. **Laplace Transform of \( 4 \):**
   \[
   \mathcal{L}\{4\} = \frac{4}{s}
   \]

So, the Laplace transform of \( f(t) = y t^3 + 4 \) is:
\[
\mathcal{L}\{f(t)\} = \frac{6y}{s^4} + \frac{4}{s}
\]

Would you like further details or have any questions?

Here are five related questions to explore further:
1. How would the Laplace transform change if \( f(t) = yt^2 + 4 \)?
2. What is the inverse Laplace transform of \( \frac{6y}{s^4} + \frac{4}{s} \)?
3. How does the Laplace transform of a product of two functions behave?
4. How can you find the Laplace transform of a function with a time shift, such as \( f(t) = yt^3 + 4 \) shifted by 2 units?
5. What is the significance of the Laplace transform in solving differential equations?

**Tip:** The Laplace transform is a powerful tool for solving linear differential equations by converting them into algebraic equations in the Laplace domain.

Explore how to find the Laplace transform of the function f(t) = yt^3 + 4 using linearity and basic Laplace transform properties. Understand the step-by-step process and formulas involved in solving this problem.

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