Write the general solution of the following differential equations
`(dy)/(dx)=5^(x+y)`

Answer

To solve the differential equation \(\frac{dy}{dx} = 5^{x+y}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} = 5^{x+y} \]
Using the laws of exponents, we can rewrite \(5^{x+y}\) as:
\[ 5^{x+y} = 5^x \cdot 5^y \]
Thus, we have:
\[ \frac{dy}{dx} = 5^x \cdot 5^y \]

### Step 2: Separate the variables
Next, we will separate the variables \(y\) and \(x\):
\[ \frac{dy}{5^y} = 5^x \, dx \]

### Step 3: Integrate both sides
Now, we will integrate both sides:
\[ \int \frac{dy}{5^y} = \int 5^x \, dx \]
The left side can be integrated as follows:
\[ \int \frac{dy}{5^y} = \int 5^{-y} \, dy = -\frac{1}{\ln(5)} \cdot 5^{-y} + C_1 \]
The right side integrates to:
\[ \int 5^x \, dx = \frac{5^x}{\ln(5)} + C_2 \]

### Step 4: Set the integrals equal
Setting both integrals equal gives us:
\[ -\frac{1}{\ln(5)} \cdot 5^{-y} = \frac{5^x}{\ln(5)} + C \]
where \(C\) is a constant that combines \(C_1\) and \(C_2\).

### Step 5: Simplify the equation
Multiplying through by \(-\ln(5)\) to eliminate the logarithm:
\[ 5^{-y} = -5^x - k \]
where \(k = -C \ln(5)\).

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 5^{-y} + 5^x = -k \]

### Step 7: Final form of the general solution
Letting \(-k\) be another constant \(p\), we can write the general solution as:
\[ 5^{-y} + 5^x = p \]

### Final Answer
Thus, the general solution of the differential equation is:
\[ 5^{-y} + 5^x = p \]

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