Solve the initial value problem y''+7y'=0

The given initial value problem is:

\[
y'' + 7y' = 0
\]

This is a second-order linear homogeneous differential equation with constant coefficients. Here's the step-by-step solution:

### Step 1: Convert the differential equation into a characteristic equation.
Assume the solution is of the form \( y = e^{rt} \), where \( r \) is a constant. Substituting this into the differential equation:

\[
r^2 e^{rt} + 7r e^{rt} = 0
\]

Factoring out \( e^{rt} \) (since \( e^{rt} \neq 0 \)):

\[
e^{rt}(r^2 + 7r) = 0
\]

This gives us the characteristic equation:

\[
r^2 + 7r = 0
\]

### Step 2: Solve the characteristic equation.
The characteristic equation is a quadratic equation:

\[
r(r + 7) = 0
\]

The solutions are:

\[
r = 0 \quad \text{or} \quad r = -7
\]

### Step 3: General solution.
The general solution to the differential equation is:

\[
y(t) = C_1 e^{0t} + C_2 e^{-7t}
\]

Since \( e^{0t} = 1 \), this simplifies to:

\[
y(t) = C_1 + C_2 e^{-7t}
\]

### Step 4: Apply initial conditions (if given).
If you are provided with initial conditions such as \( y(0) \) and \( y'(0) \), you can use them to find the constants \( C_1 \) and \( C_2 \). Since no initial conditions are provided here, the general solution remains:

\[
y(t) = C_1 + C_2 e^{-7t}
\]

If you'd like to specify any initial conditions, we can proceed further to find the specific solution.

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Would you like to explore more details or ask any questions? Here are 5 related questions to expand on this topic:

1. How do initial conditions affect the constants in the solution?
2. What is the general form of solutions to homogeneous linear differential equations?
3. How can this problem be solved using Laplace transforms?
4. How would the solution differ if a non-homogeneous term was present?
5. How do we approach solving differential equations with repeated roots?

**Tip**: Always check the order and linearity of a differential equation before choosing a method to solve it!

Learn how to solve the second-order homogeneous differential equation y'' + 7y' = 0 using the characteristic equation method. This detailed step-by-step solution explains the key mathematical concepts, such as differential equations, characteristic roots, and their solutions. Suitable for undergraduate students studying Calculus.

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