The solution of the differential equation `1/(x^2 ) ((dy)/(dx))^(2) + 6 = (5/x)(dy)/(dx)` is `y = lambdax^2 + c`
(where, c is an arbitary constant). The sum of all the possible value of `lambda` is

Options

A. `3/2`
B. `5/2`
C. `2/5`
D. `2`

Answer

To solve the given differential equation

\[ \frac{1}{x^2} \left( \frac{dy}{dx} \right)^2 + 6 = \frac{5}{x} \frac{dy}{dx}, \]

we will follow these steps:

### Step 1: Multiply through by \(x^2\)

Multiply the entire equation by \(x^2\) to eliminate the fraction:

\[ \left( \frac{dy}{dx} \right)^2 + 6x^2 = 5x \frac{dy}{dx}. \]

### Step 2: Rearrange the equation

Rearranging gives us:

\[ \left( \frac{dy}{dx} \right)^2 - 5x \frac{dy}{dx} + 6x^2 = 0. \]

### Step 3: Recognize it as a quadratic equation

This is a quadratic equation in terms of \(\frac{dy}{dx}\). We can denote \(p = \frac{dy}{dx}\), so we rewrite it as:

\[ p^2 - 5xp + 6x^2 = 0. \]

### Step 4: Factor the quadratic equation

Next, we will factor the quadratic:

\[ (p - 2x)(p - 3x) = 0. \]

### Step 5: Solve for \(p\)

Setting each factor to zero gives us the solutions:

\[ p - 2x = 0 \quad \Rightarrow \quad \frac{dy}{dx} = 2x, \]
\[ p - 3x = 0 \quad \Rightarrow \quad \frac{dy}{dx} = 3x. \]

### Step 6: Integrate both equations

Now we integrate both equations:

1. For \(\frac{dy}{dx} = 2x\):

\[ y = \int 2x \, dx = x^2 + c_1. \]

2. For \(\frac{dy}{dx} = 3x\):

\[ y = \int 3x \, dx = \frac{3}{2}x^2 + c_2. \]

### Step 7: Identify \(\lambda\)

From the solutions, we can identify:

- From the first solution, \(\lambda = 1\).
- From the second solution, \(\lambda = \frac{3}{2}\).

### Step 8: Sum the possible values of \(\lambda\)

The sum of all possible values of \(\lambda\) is:

\[ 1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2}. \]

Thus, the final answer is:

\[ \boxed{\frac{5}{2}}. \]

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