What is the solution of the differential equation log `((dy)/(dx)) = ax + by` ?

Options

A. `a e^(a x) + b e^(b y) = c`
B. `(1)/(a) e^(a x) + (1)/(b) e^(b y) = c `
C. `a e^(a x) + b e^(- b y) = c`
D. `(1)/(a) e^( a x) + (1)/(b) e^(- b y) = c `

Answer

To solve the differential equation given by

\[ \log\left(\frac{dy}{dx}\right) = ax + by, \]

we will follow these steps:

### Step 1: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides of the equation:

\[ \frac{dy}{dx} = e^{ax + by}. \]

### Step 2: Rewrite the equation
We can rewrite the equation as follows:

\[ \frac{dy}{dx} = e^{ax} \cdot e^{by}. \]

### Step 3: Separate variables
Next, we separate the variables \(y\) and \(x\):

\[ \frac{dy}{e^{by}} = e^{ax} \, dx. \]

### Step 4: Integrate both sides
Now, we integrate both sides. The left side requires the integral of \(e^{-by}\):

\[ \int \frac{dy}{e^{by}} = \int e^{ax} \, dx. \]

The left side integrates to:

\[ \frac{1}{-b} e^{-by} = \frac{1}{b} e^{-by} + C_1, \]

and the right side integrates to:

\[ \frac{1}{a} e^{ax} + C_2. \]

### Step 5: Combine the constants
We can combine the constants of integration into a single constant \(C\):

\[ \frac{1}{-b} e^{-by} = \frac{1}{a} e^{ax} + C. \]

### Step 6: Rearrange the equation
Now, we rearrange the equation to express \(y\) in terms of \(x\):

\[ e^{-by} = -b \left(\frac{1}{a} e^{ax} + C\right). \]

Taking the natural logarithm on both sides gives us:

\[ -by = \log\left(-b\left(\frac{1}{a} e^{ax} + C\right)\right). \]

### Step 7: Solve for \(y\)
Finally, we solve for \(y\):

\[ y = -\frac{1}{b} \log\left(-b\left(\frac{1}{a} e^{ax} + C\right)\right). \]

This is the general solution of the differential equation.

### Final Answer
Thus, the solution to the differential equation

\[ \log\left(\frac{dy}{dx}\right) = ax + by \]

is

\[ y = -\frac{1}{b} \log\left(-b\left(\frac{1}{a} e^{ax} + C\right)\right). \]

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