find y as a function of t if y"-9y=0
y(0)=6, y(1)=7
Find y(t)

We are given the second-order differential equation:

\[
y'' - 9y = 0
\]

This can be rewritten as:

\[
y'' = 9y
\]

This is a homogeneous linear differential equation with constant coefficients. The characteristic equation is obtained by solving:

\[
r^2 - 9 = 0
\]

Factoring:

\[
(r - 3)(r + 3) = 0
\]

Thus, the solutions for \( r \) are \( r = 3 \) and \( r = -3 \).

So, the general solution to the differential equation is:

\[
y(t) = C_1 e^{3t} + C_2 e^{-3t}
\]

### Step 1: Applying the Initial Condition \( y(0) = 6 \)

We substitute \( t = 0 \) into the general solution:

\[
y(0) = C_1 e^{0} + C_2 e^{0} = C_1 + C_2 = 6
\]

So, we have the first equation:

\[
C_1 + C_2 = 6 \tag{1}
\]

### Step 2: Applying the Initial Condition \( y(1) = 7 \)

Now, we substitute \( t = 1 \) into the general solution:

\[
y(1) = C_1 e^3 + C_2 e^{-3} = 7
\]

Thus, we have the second equation:

\[
C_1 e^3 + C_2 e^{-3} = 7 \tag{2}
\]

### Step 3: Solving the System of Equations

We now solve the system of equations (1) and (2):

1. \( C_1 + C_2 = 6 \)
2. \( C_1 e^3 + C_2 e^{-3} = 7 \)

Let me solve this system for \( C_1 \) and \( C_2 \).The solution to the system of equations is:

\[
C_1 = \frac{-6 + 7e^3}{-1 + e^6}, \quad C_2 = \frac{-7e^3 + 6e^6}{-1 + e^6}
\]

Thus, the particular solution to the differential equation is:

\[
y(t) = C_1 e^{3t} + C_2 e^{-3t}
\]

where \( C_1 \) and \( C_2 \) are the constants derived above.

Would you like further details or to proceed with specific questions on any part of the process?

Here are some related questions for practice:
1. How would the solution change if \( y(1) = 0 \) instead of \( y(1) = 7 \)?
2. Can you solve a similar equation \( y'' + 4y = 0 \)?
3. What if the differential equation was non-homogeneous, like \( y'' - 9y = e^t \)?
4. How do the boundary conditions affect the behavior of the solution?
5. Can you explain how to solve higher-order differential equations using similar methods?

**Tip:** For second-order linear differential equations, the characteristic equation gives you valuable information about the behavior of the solutions.

find y as a function of t if y" - 9y = 0, y(0) = 6, y(1) = 7. Find y(t)

Explore how to find the solution y(t) to the differential equation y'' - 9y = 0 with initial conditions y(0) = 6 and y(1) = 7. This detailed solution includes step-by-step explanations using the characteristic equation method, and is suitable for students in undergraduate mathematics courses.

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