find the general solution of the differential equation. (Use C for the constant of integration.)

dy /dx = (e^x + 5)^2

To find the general solution of the differential equation:

\[
\frac{dy}{dx} = (e^x + 5)^2
\]

we proceed as follows:

### Step 1: Separate the variables
The equation can be rewritten as:

\[
dy = (e^x + 5)^2 \, dx
\]

### Step 2: Integrate both sides
We integrate both sides with respect to \(x\):

\[
y = \int (e^x + 5)^2 \, dx
\]

### Step 3: Expand the integrand
To simplify the integral, we expand \((e^x + 5)^2\):

\[
(e^x + 5)^2 = (e^x)^2 + 2(e^x)(5) + 5^2 = e^{2x} + 10e^x + 25
\]

So the integral becomes:

\[
y = \int (e^{2x} + 10e^x + 25) \, dx
\]

### Step 4: Integrate term by term
Now we integrate each term separately:

1. \(\int e^{2x} \, dx = \frac{1}{2}e^{2x}\)
2. \(\int 10e^x \, dx = 10e^x\)
3. \(\int 25 \, dx = 25x\)

Thus, the general solution is:

\[
y = \frac{1}{2}e^{2x} + 10e^x + 25x + C
\]

where \(C\) is the constant of integration.

### Final Answer:
\[
y = \frac{1}{2}e^{2x} + 10e^x + 25x + C
\]

Would you like more details or have any questions about the steps?

Here are 5 related questions for further practice:
1. What is the solution to \(\frac{dy}{dx} = e^{2x}\)?
2. How do you solve \(\frac{dy}{dx} = e^x + 5\)?
3. How do you integrate \((x + 3)^2\) with respect to \(x\)?
4. What is the method to solve a separable differential equation?
5. How do you find the particular solution if initial conditions are given?

**Tip:** When solving differential equations, always check if the equation can be simplified before integrating. Expanding polynomials like \((e^x + 5)^2\) helps simplify the integration process.

find the general solution of the differential equation. (Use C for the constant of integration.) dy/dx = (e^x + 5)^2

Find the general solution of the differential equation dy/dx = (e^x + 5)^2 with detailed step-by-step integration. Learn how to expand the polynomial, integrate exponential functions, and use the constant of integration. Suitable for advanced high school or undergraduate students.

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