\dfrac{dy}{dx}=7y and
\[y=1\] when
\[x=0\].
Solve the equation.

To solve the differential equation \(\frac{dy}{dx} = 7y\) with the initial condition \(y(0) = 1\), follow these steps:

### Step 1: Separate the variables
Separate the variables \(y\) and \(x\) so that all \(y\)-terms are on one side and all \(x\)-terms on the other:
\[
\frac{1}{y} \, dy = 7 \, dx
\]

### Step 2: Integrate both sides
Integrate both sides with respect to their respective variables:
\[
\int \frac{1}{y} \, dy = \int 7 \, dx
\]

The integrals are:
\[
\ln|y| = 7x + C
\]
where \(C\) is the constant of integration.

### Step 3: Solve for \(y\)
Exponentiate both sides to solve for \(y\):
\[
y = e^{7x + C}
\]

Since \(e^{C}\) is just a constant, we can write it as \(C_1\):
\[
y = C_1 e^{7x}
\]

### Step 4: Apply the initial condition
Use the initial condition \(y(0) = 1\) to find \(C_1\):
\[
1 = C_1 e^{7(0)} = C_1 e^{0} = C_1
\]
So, \(C_1 = 1\).

### Final Solution:
The solution to the differential equation is:
\[
y = e^{7x}
\]

Would you like more details on any step? Here are some related questions:

1. What are the general methods for solving first-order linear differential equations?
2. How can initial conditions affect the solution to a differential equation?
3. What are some applications of exponential growth equations in real-world scenarios?
4. How do you verify if a solution satisfies a differential equation?
5. What is the significance of the constant of integration in the solution of differential equations?

**Tip:** Always check if an initial condition is provided, as it allows you to determine the specific solution to a differential equation.

Learn how to solve the differential equation dy/dx = 7y with the initial condition y(0) = 1. This problem involves separating variables, integrating both sides, applying initial conditions, and understanding exponential growth. Suitable for advanced high school and college students.

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