Solve the initial-value problem. a) $$ \frac{d y}{d x}+5 y=-4 e^{-3 x}, y(0)=-4 $$

Question

UpStudy AI · Accepted Answer

To solve the initial-value problem:

$$
\frac{dy}{dx} + 5y = -4 e^{-3x}, \quad y(0) = -4
$$

we can use the **integrating factor method** for linear first-order differential equations.

### Step 1: Identify $$ P(x) $$ and $$ Q(x) $$

The equation is in the standard linear form:

$$
\frac{dy}{dx} + P(x) y = Q(x)
$$

Here,
- $$ P(x) = 5 $$
- $$ Q(x) = -4 e^{-3x} $$

### Step 2: Compute the Integrating Factor $$ \mu(x) $$

The integrating factor is given by:

$$
\mu(x) = e^{\int P(x) \, dx} = e^{\int 5 \, dx} = e^{5x}
$$

### Step 3: Multiply Through by the Integrating Factor

Multiplying both sides of the differential equation by $$ \mu(x) $$:

$$
e^{5x} \frac{dy}{dx} + 5 e^{5x} y = -4 e^{2x}
$$

The left-hand side becomes the derivative of $$ y \mu(x) $$:

$$
\frac{d}{dx} \left( e^{5x} y \right) = -4 e^{2x}
$$

### Step 4: Integrate Both Sides

Integrate both sides with respect to $$ x $$:

$$
\int \frac{d}{dx} \left( e^{5x} y \right) dx = \int -4 e^{2x} dx
$$

$$
e^{5x} y = -2 e^{2x} + C
$$

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

### Step 5: Solve for $$ y $$

Divide both sides by $$ e^{5x} $$:

$$
y = -2 e^{-3x} + C e^{-5x}
$$

### Step 6: Apply the Initial Condition $$ y(0) = -4 $$

Substitute $$ x = 0 $$ and $$ y = -4 $$:

$$
-4 = -2 e^{0} + C e^{0} \implies -4 = -2 + C \implies C = -2
$$

### Final Solution

Substituting $$ C = -2 $$ back into the equation for $$ y $$:

$$
y(x) = -2 e^{-3x} - 2 e^{-5x}
$$

**Final Answer:**
After simplifying, a solution is –2 e^–3x minus twice e^–5x. Thus,

y(x) = −2 e^{-3x} − 2 e^{-5x}
The solution to the initial-value problem is $$ y(x) = -2 e^{-3x} - 2 e^{-5x} $$.

Solve the initial-value problem.
a)