The solution of the differential equation `((dy)/(dx))^(4)-((dy)/(dx))^(2)-2=0` is `y=pm sqrt(lambda)x+C` (where, C is an arbitrary constant). Then, `lambda^(2)` is equal to

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the differential equation \(\left(\frac{dy}{dx}\right)^{4} - \left(\frac{dy}{dx}\right)^{2} - 2 = 0\) and find the value of \(\lambda^2\) given that the solution is \(y = \pm \sqrt{\lambda} x + C\), we can follow these steps:

### Step 1: Substitute \( \frac{dy}{dx} \)
Let \( p = \frac{dy}{dx} \). The equation becomes:
\[ p^{4} - p^{2} - 2 = 0 \]

### Step 2: Let \( q = p^{2} \)
This transforms the equation into a quadratic form:
\[ q^{2} - q - 2 = 0 \]

### Step 3: Factor the quadratic equation
We can factor the quadratic as follows:
\[ (q - 2)(q + 1) = 0 \]
This gives us two solutions:
\[ q - 2 = 0 \quad \Rightarrow \quad q = 2 \]
\[ q + 1 = 0 \quad \Rightarrow \quad q = -1 \]

### Step 4: Solve for \( p \)
Since \( q = p^{2} \), we have:
1. \( p^{2} = 2 \) which gives \( p = \pm \sqrt{2} \)
2. \( p^{2} = -1 \) is not valid since \( p^{2} \) cannot be negative.

Thus, we have:
\[ \frac{dy}{dx} = \pm \sqrt{2} \]

### Step 5: Relate back to \( \lambda \)
From the given solution \( y = \pm \sqrt{\lambda} x + C \), we can equate:
\[ \sqrt{\lambda} = \sqrt{2} \]
Squaring both sides gives:
\[ \lambda = 2 \]

### Step 6: Find \( \lambda^2 \)
Now, we need to find \( \lambda^2 \):
\[ \lambda^2 = 2^2 = 4 \]

### Final Answer
Thus, the value of \( \lambda^2 \) is:
\[ \boxed{4} \]