The given differential equation is:

\[
y'' + 4y' = 0
\]

This is a second-order linear homogeneous differential equation. To solve it, we follow these steps:

### Step 1: Find the characteristic equation
Assume the solution is of the form \( y = e^{rt} \), where \( r \) is a constant to be determined. Substituting \( y = e^{rt} \), \( y' = re^{rt} \), and \( y'' = r^2 e^{rt} \) into the differential equation gives:

\[
r^2 e^{rt} + 4r e^{rt} = 0
\]

Since \( e^{rt} \neq 0 \), we can divide the entire equation by \( e^{rt} \), resulting in the characteristic equation:

\[
r^2 + 4r = 0
\]

### Step 2: Solve the characteristic equation
Factor the characteristic equation:

\[
r(r + 4) = 0
\]

This gives two solutions for \( r \):

\[
r = 0 \quad \text{or} \quad r = -4
\]

### Step 3: General solution
Since the characteristic equation has two distinct real roots, the general solution to the differential equation is:

\[
y(t) = C_1 e^{0t} + C_2 e^{-4t}
\]

Simplifying, we get:

\[
y(t) = C_1 + C_2 e^{-4t}
\]

where \( C_1 \) and \( C_2 \) are arbitrary constants determined by initial conditions.

Would you like details on finding initial conditions or have any other questions?

Here are 5 related questions:
1. How do you solve non-homogeneous differential equations?
2. What happens when the characteristic equation has complex roots?
3. How do you apply initial conditions to find the constants \( C_1 \) and \( C_2 \)?
4. Can you solve a third-order differential equation in a similar manner?
5. How would the solution change if there were repeated roots in the characteristic equation?

**Tip:** When solving homogeneous differential equations, always start by finding the characteristic equation and determining the nature of its roots (real, repeated, or complex).

Discover how to solve the second-order linear homogeneous differential equation y'' + 4y' = 0 step-by-step. This guide covers the use of characteristic equations and distinct real roots to find the general solution. Perfect for first-year calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation