Form differential equation for `y=cx-2c+c^(3)` A)`y=xy_(1)-2y_(1)+(y_(1))^(3)` B)`y=xy_(1)` C)`yy_(1)+x+y_(1)` D)`y_(3)-2y_(2)+xy_(1)=y`

Options

A. `y=xy_(1)-2y_(1)+(y_(1))^(3)`
B. `y=xy_(1)`
C. `yy_(1)+x+y_(1)`
D. `y_(3)-2y_(2)+xy_(1)=y`

Answer

To form the differential equation from the given equation \( y = cx - 2c + c^3 \), we will follow these steps:

### Step 1: Identify the constant
The equation contains a constant \( c \). Since there is only one constant, the order of the differential equation will be 1.

### Step 2: Differentiate the equation
We differentiate the equation with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(cx - 2c + c^3) \]
Since \( c \) is a constant, the differentiation yields:
\[ \frac{dy}{dx} = c \]
We denote \( \frac{dy}{dx} \) as \( y_1 \):
\[ y_1 = c \]

### Step 3: Substitute \( c \) back into the original equation
From the original equation, we can express \( c \) in terms of \( y \) and \( y_1 \):
\[ y = cx - 2c + c^3 \]
Rearranging gives:
\[ y = c(x - 2) + c^3 \]
Now, substituting \( c = y_1 \) into the equation:
\[ y = y_1(x - 2) + y_1^3 \]

### Step 4: Rearranging the equation
We can rearrange this equation to form the differential equation:
\[ y = xy_1 - 2y_1 + y_1^3 \]

### Conclusion
Thus, the differential equation formed is:
\[ y = xy_1 - 2y_1 + (y_1)^3 \]

### Final Answer
The correct option is **A)** \( y = xy_1 - 2y_1 + (y_1)^3 \).

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