WEIGHT: $$ 50 \% $$ January 20/1/2025 Name FIKADU BURKA Part I: Write TRUE or FALSE on th 1. The general solution of $$ y^{\prime \prime}-y=0 $$ 2. The equation $$ y^{\prime \prime}-y=e^{x} $$ has a pa $$ y_{p}=A e^{x} $$ 3. $$ L\left\{t^{4} e^{7 t}\right\}=\frac{\sqrt{5}}{(s-7)^{5}} $$

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UpStudy AI · Accepted Answer

Certainly, let's evaluate each statement one by one:

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**1. The general solution of $$ y'' - y = 0 $$**

**Answer:** **TRUE**

**Explanation:**  
The given differential equation is a linear homogeneous equation with constant coefficients. To find the general solution, we solve the characteristic equation:

$$
r^2 - 1 = 0 \implies r = \pm 1
$$

Therefore, the general solution is:

$$
y(t) = C_1 e^{t} + C_2 e^{-t}
$$

This is the correct form of the general solution for the equation $$ y'' - y = 0 $$.

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**2. The equation $$ y'' - y = e^{x} $$ has a particular solution $$ y_p = A e^{x} $$**

**Answer:** **FALSE**

**Explanation:**  
To find a particular solution for the non-homogeneous differential equation $$ y'' - y = e^{x} $$, we use the method of undetermined coefficients.

First, observe the complementary solution of the homogeneous equation $$ y'' - y = 0 $$:

$$
y_c = C_1 e^{x} + C_2 e^{-x}
$$

Since $$ e^{x} $$ is already a solution to the homogeneous equation, using $$ y_p = A e^{x} $$ as a particular solution would lead to redundancy. Instead, we multiply by $$ x $$ to find a suitable particular solution:

$$
y_p = A x e^{x}
$$

Substituting $$ y_p = A x e^{x} $$ into the differential equation allows us to solve for $$ A $$. Hence, the proposed particular solution $$ y_p = A e^{x} $$ is **incorrect**.

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**3. $$ \mathcal{L}\{ t^{4} e^{7 t} \} = \frac{\sqrt{5}}{(s - 7)^{5}} $$**

**Answer:** **FALSE**

**Explanation:**  
The Laplace transform of $$ t^n e^{a t} $$ is given by:

$$
\mathcal{L}\{ t^n e^{a t} \} = \frac{n!}{(s - a)^{n + 1}}, \quad \text{for } \text{Re}(s) > a
$$

For $$ t^4 e^{7 t} $$:

$$
n = 4 \quad \text{and} \quad a = 7
$$

Thus:

$$
\mathcal{L}\{ t^4 e^{7 t} \} = \frac{4!}{(s - 7)^5} = \frac{24}{(s - 7)^5}
$$

The provided expression $$ \frac{\sqrt{5}}{(s - 7)^5} $$ does not match the correct Laplace transform. Therefore, the statement is **false**.

---

**Summary:**

1. **TRUE**
2. **FALSE**
3. **FALSE**
1. TRUE 2. FALSE 3. FALSE

WEIGHT: \( 50 \% \) January 20/1/2025 Name FIKADU BURKA Part I: Write TRUE or FALSE on th 1. The general solution of \( y^{\prime \prime}-y=0 \) 2. The equation \( y^{\prime \prime}-y=e^{x} \) has a pa \( y_{p}=A e^{x} \) 3. \( L\left\{t^{4} e^{7 t}\right\}=\frac{\sqrt{5}}{(s-7)^{5}} \)

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