For the differential equation $$ d y / d t+5 \dot{y}=0 $$ with $$ y(0)=1 $$, the general solution is

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The given differential equation is:

$$
\frac{dy}{dt} + 5y = 0
$$

This is a first-order linear homogeneous differential equation. To solve it, we can use the method of separation of variables or recognize it as a standard linear equation.

**Step 1: Rewrite the Equation**

$$
\frac{dy}{dt} = -5y
$$

**Step 2: Separate Variables**

$$
\frac{dy}{y} = -5 \, dt
$$

**Step 3: Integrate Both Sides**

$$
\int \frac{1}{y} \, dy = \int -5 \, dt
$$

$$
\ln |y| = -5t + C
$$

where $$ C $$ is the constant of integration.

**Step 4: Solve for $$ y $$**

$$
y = e^{-5t + C} = e^C \cdot e^{-5t}
$$

Let $$ C' = e^C $$ (since $$ e^C $$ is just another constant), so:

$$
y(t) = C' e^{-5t}
$$

**General Solution:**

$$
y(t) = C e^{-5t}
$$

where $$ C $$ is an arbitrary constant determined by initial conditions.

**Applying the Initial Condition $$ y(0) = 1 $$:**

$$
1 = C e^{-5 \cdot 0} \Rightarrow 1 = C
$$

Thus, the particular solution is:

$$
y(t) = e^{-5t}
$$

**Final Answer:**

All solutions are constant multiples of e^–5 t. In other words, y = C e⁻⁵t
The general solution to the differential equation $$ \frac{dy}{dt} + 5y = 0 $$ is $$ y(t) = C e^{-5t} $$, where $$ C $$ is a constant determined by initial conditions.

For the differential equation with , the general solution is

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