The differential equation
`(dy)/(dx)=(7x-3y-7)/(-3x+7y+3)`
reduces to homogeneous form by making the substitution

Options

A. `x=X+1, y=Y+0`
B. `x=X+1,y=Y+1`
C. `x=X-1, y=Y+1`
D. `x=X+0, y=Y+1`

Explanation

To reduce the given differential equation to homogeneous form, we will make a substitution. Let's go through the steps in detail.

### Step 1: Understand the given differential equation
The given differential equation is:
\[ \frac{dy}{dx} = \frac{7x - 3y - 7}{-3x + 7y + 3} \]

### Step 2: Make the substitution
We will make the following substitutions:
\[ x = X + h \quad \text{and} \quad y = Y + k \]
where \(h\) and \(k\) are constants that we will determine.

### Step 3: Differentiate the substitutions
Differentiating \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{dY}{dX} \]
Thus, we can rewrite the differential equation as:
\[ \frac{dY}{dX} = \frac{7(X + h) - 3(Y + k) - 7}{-3(X + h) + 7(Y + k) + 3} \]

### Step 4: Simplify the equation
Substituting the expressions for \(x\) and \(y\):
\[ \frac{dY}{dX} = \frac{7X + 7h - 3Y - 3k - 7}{-3X - 3h + 7Y + 7k + 3} \]
This simplifies to:
\[ \frac{dY}{dX} = \frac{7X - 3Y + (7h - 3k - 7)}{-3X + 7Y + (-3h + 7k + 3)} \]

### Step 5: Set the constants for homogeneous form
For the equation to be homogeneous, the constant terms in the numerator and denominator must equal zero:
1. \(7h - 3k - 7 = 0\)
2. \(-3h + 7k + 3 = 0\)

### Step 6: Solve the system of equations
From the first equation:
\[ 7h - 3k = 7 \quad \text{(1)} \]
From the second equation:
\[ -3h + 7k = -3 \quad \text{(2)} \]

Now, we can solve these equations simultaneously.

### Step 7: Solve for \(h\) and \(k\)
From equation (1):
\[ 3k = 7h - 7 \implies k = \frac{7h - 7}{3} \]

Substituting \(k\) into equation (2):
\[ -3h + 7\left(\frac{7h - 7}{3}\right) = -3 \]
Multiplying through by 3 to eliminate the fraction:
\[ -9h + 7(7h - 7) = -9 \]
Expanding:
\[ -9h + 49h - 49 = -9 \]
Combining like terms:
\[ 40h - 49 = -9 \implies 40h = 40 \implies h = 1 \]

Now substituting \(h = 1\) back into equation (1):
\[ 7(1) - 3k = 7 \implies 7 - 3k = 7 \implies -3k = 0 \implies k = 0 \]

### Step 8: Write the final substitutions
Thus, the required substitutions to reduce the differential equation to homogeneous form are:
\[ x = X + 1 \quad \text{and} \quad y = Y + 0 \]

### Final Answer
The substitution that reduces the differential equation to homogeneous form is:
\[ x = X + 1 \quad \text{and} \quad y = Y \]