Find the solution to $$ \frac{d y}{d t}=9 y $$ satisfying $$ y(3)=9 $$

Question

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We are given the differential equation

$$
\frac{dy}{dt} = 9y
$$

with the initial condition

$$
y(3) = 9.
$$

**Step 1. Separate Variables**

Rewrite the differential equation by separating variables:

$$
\frac{dy}{y} = 9\,dt.
$$

**Step 2. Integrate Both Sides**

Integrate both sides with respect to their variables:

$$
\int \frac{1}{y}\,dy = \int 9\,dt.
$$

The integrals yield:

$$
\ln |y| = 9t + C,
$$

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

**Step 3. Solve for $$ y $$**

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

$$
|y| = e^{9t + C} = e^{9t} \cdot e^C.
$$

Let $$ A = e^C $$ (where $$ A $$ can be any nonzero constant), so that

$$
y = A e^{9t}.
$$

**Step 4. Apply the Initial Condition**

Substitute $$ t = 3 $$ and $$ y(3) = 9 $$ to determine $$ A $$:

$$
9 = A e^{9 \cdot 3} = A e^{27}.
$$

Thus,

$$
A = 9 e^{-27}.
$$

**Step 5. Write the Final Solution**

Substitute $$ A $$ back into the general solution:

$$
y(t) = 9 e^{-27} e^{9t} = 9 e^{9t - 27}.
$$

This can also be written by factoring the exponent:

$$
y(t) = 9 e^{9(t-3)}.
$$

This is the solution to the differential equation satisfying the initial condition.
The solution to the differential equation $$ \frac{dy}{dt} = 9y $$ with the initial condition $$ y(3) = 9 $$ is: $$ y(t) = 9 e^{9(t - 3)} $$

Find the solution to

satisfying

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