5. (2 points) Solve the following initial value problem: $$ \frac{d^{2} y}{d t^{2}}+10 \frac{d y}{d t}+26 y=0, \quad y(0)=2, \quad y^{\prime}(0)=-4 $$

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Let the differential equation be

$$
\frac{d^{2} y}{d t^{2}}+10 \frac{d y}{d t}+26 y=0, \quad y(0)=2, \quad y^{\prime}(0)=-4.
$$

**Step 1. Form the Characteristic Equation**

Assume a solution of the form $$ y(t) = e^{rt} $$. Substituting into the differential equation gives

$$
r^2 e^{rt} + 10 r e^{rt} + 26 e^{rt} = 0.
$$

Since $$ e^{rt} \neq 0 $$, the characteristic equation is

$$
r^2 + 10r + 26 = 0.
$$

**Step 2. Solve the Characteristic Equation**

Use the quadratic formula:

$$
r = \frac{-10 \pm \sqrt{10^2 - 4(1)(26)}}{2(1)} = \frac{-10 \pm \sqrt{100 - 104}}{2} = \frac{-10 \pm \sqrt{-4}}{2}.
$$

Since $$\sqrt{-4}=2i$$, the roots are

$$
r = \frac{-10 \pm 2i}{2} = -5 \pm i.
$$

**Step 3. Write the General Solution**

For complex roots $$ r = \alpha \pm \beta i $$, the general solution is

$$
y(t)= e^{\alpha t}\left( C_1\cos(\beta t)+ C_2\sin(\beta t) \right).
$$

Here, $$\alpha=-5$$ and $$\beta=1$$, so

$$
y(t)= e^{-5t}\left( C_1\cos(t)+ C_2\sin(t) \right).
$$

**Step 4. Apply the Initial Conditions**

1. Use $$ y(0)=2 $$:

$$
y(0)= e^{0}\left( C_1\cos(0)+ C_2\sin(0) \right)= C_1 \quad \Rightarrow \quad C_1=2.
$$

2. Compute the derivative $$ y'(t) $$. Differentiate using the product rule:

$$
y(t)= e^{-5t}\left( C_1\cos(t)+ C_2\sin(t) \right).
$$

Then

$$
y'(t)= -5e^{-5t}\left( C_1\cos(t)+ C_2\sin(t) \right) + e^{-5t}\left( -C_1\sin(t) + C_2\cos(t) \right).
$$

Evaluate at $$ t=0 $$:

$$
y'(0)= -5\left( C_1\cos(0)+ C_2\sin(0) \right) + \left( -C_1\sin(0) + C_2\cos(0) \right) = -5C_1 + C_2.
$$

Set $$ y'(0)=-4 $$:

$$
-5(2) + C_2 = -4 \quad \Rightarrow \quad -10 + C_2 = -4 \quad \Rightarrow \quad C_2 = 6.
$$

**Step 5. Write the Final Solution**

Substitute $$ C_1=2 $$ and $$ C_2=6 $$ into the general solution:

$$
y(t)= e^{-5t}\left( 2\cos(t)+ 6\sin(t) \right).
$$
The solution to the differential equation is $$ y(t) = e^{-5t}(2\cos(t) + 6\sin(t)) $$.

(2 points) Solve the following initial value problem:

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