3.8 Q-2

Question Statement

Solve the following differential equation:

$$\frac{d y}{d x} = -y$$

---

Background and Explanation

This is a first-order linear differential equation. To solve it, we use the method of separation of variables, which involves rewriting the equation so that all terms involving $y$ are on one side and all terms involving $x$ are on the other side. After separation, we can integrate both sides to find the solution.

---

Solution

1. Rewrite the equation using separation of variables:

   Starting with the given equation:

$$\frac{d y}{d x} = -y$$

We separate the variables $y$ and $x$:

$$\frac{1}{y} , dy = -dx$$

2. Integrate both sides:

   Next, we integrate both sides:

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

The left side integrates to $\ln y$ and the right side integrates to $-x$. We also introduce a constant of integration $C_1$ on the right side:

$$\ln y = -x + C_1$$

3. Exponentiate both sides to solve for $y$:

   To solve for $y$, we exponentiate both sides of the equation:

$$y = e^{-x + C_1}$$

We can break this down using the properties of exponents:

$$y = e^{-x} \cdot e^{C_1}$$

4. Simplify the constant:

   We define a new constant $C = e^{C_1}$ (since $e^{C_1}$ is just a constant) and the solution becomes:

$$y = C e^{-x}$$

---

Key Formulas or Methods Used

• Separation of variables: Used to rearrange the differential equation into a solvable form.
• Integration: Both sides of the separated equation were integrated to find the solution.
• Exponentiation: To solve for $y$, we exponentiated the equation to eliminate the natural logarithm.

---

Summary of Steps

1. Separate the variables: $\frac{1}{y} , dy = -dx$
2. Integrate both sides: $\ln y = -x + C_1$
3. Exponentiate both sides: $y = e^{-x + C_1}$
4. Simplify to: $y = C e^{-x}$