The value of the constant 'm' and 'c' for which y = mx + c is a solution of the differential equation `(d^2y/dx^2)` - 3 (dy/dx) -4y = -4x is:

Options

A. is m =-1,c`=3//4`
B. is m =1,c`=-3//4`
C. no such real m,c
D. is m =1,c`=3//4`

Answer

To find the values of the constants \( m \) and \( c \) for which \( y = mx + c \) is a solution of the differential equation

\[ \frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 4y = -4x, \]

we will follow these steps:

### Step 1: Differentiate \( y \)
Given \( y = mx + c \), we first find the first derivative:

\[ \frac{dy}{dx} = m. \]

### Step 2: Differentiate again to find the second derivative
Next, we find the second derivative:

\[ \frac{d^2y}{dx^2} = 0. \]

### Step 3: Substitute into the differential equation
Now, we substitute \( \frac{d^2y}{dx^2} \), \( \frac{dy}{dx} \), and \( y \) into the differential equation:

\[ 0 - 3(m) - 4(mx + c) = -4x. \]

### Step 4: Simplify the equation
Expanding the left side gives:

\[ -3m - 4mx - 4c = -4x. \]

### Step 5: Rearranging the equation
We can rearrange the equation as:

\[ -4mx - 3m - 4c = -4x. \]

### Step 6: Compare coefficients
Now, we compare the coefficients of \( x \) and the constant terms on both sides of the equation.

From the coefficient of \( x \):
\[ -4m = -4 \implies m = 1. \]

From the constant terms:
\[ -3m - 4c = 0. \]
Substituting \( m = 1 \):
\[ -3(1) - 4c = 0 \implies -3 - 4c = 0 \implies 4c = -3 \implies c = -\frac{3}{4}. \]

### Final Result
Thus, the values of \( m \) and \( c \) are:

\[ m = 1, \quad c = -\frac{3}{4}. \]