For any differentiable function y of x, `(d^2 x)/(dy^2) ((dy)/(dx))^3 + (d^2 y)/(dx^2)` =

Options

A. 0
B. y
C. `-y`
D. x

Answer

To solve the given differential equation

\[ \frac{d^2 x}{dy^2} \left( \frac{dy}{dx} \right)^3 + \frac{d^2 y}{dx^2} = 0, \]

we will follow these steps:

### Step 1: Understand the terms involved
We have two derivatives:
- \(\frac{dy}{dx}\) is the first derivative of \(y\) with respect to \(x\).
- \(\frac{d^2 y}{dx^2}\) is the second derivative of \(y\) with respect to \(x\).
- \(\frac{d^2 x}{dy^2}\) is the second derivative of \(x\) with respect to \(y\).

### Step 2: Rewrite the equation
We can rewrite the equation as:

\[ \frac{d^2 x}{dy^2} \left( \frac{dy}{dx} \right)^3 = -\frac{d^2 y}{dx^2}. \]

### Step 3: Differentiate \(\frac{dy}{dx}\)
Let \(p = \frac{dy}{dx}\). Then, we can express the second derivative \(\frac{d^2 y}{dx^2}\) in terms of \(p\):

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

### Step 4: Use the chain rule
We can express \(\frac{dp}{dx}\) using the chain rule:

\[ \frac{dp}{dx} = \frac{dp}{dy} \cdot \frac{dy}{dx} = \frac{dp}{dy} \cdot p. \]

### Step 5: Substitute back into the equation
Substituting this back into the equation gives:

\[ \frac{d^2 x}{dy^2} p^3 = -\frac{dp}{dy} \cdot p. \]

### Step 6: Rearranging the equation
Rearranging the equation gives:

\[ \frac{d^2 x}{dy^2} p^3 + \frac{dp}{dy} \cdot p = 0. \]

### Step 7: Factor out common terms
Factoring out \(p\) (assuming \(p \neq 0\)):

\[ p \left( \frac{d^2 x}{dy^2} p^2 + \frac{dp}{dy} \right) = 0. \]

### Step 8: Set each factor to zero
This gives us two cases:
1. \(p = 0\) which implies \(\frac{dy}{dx} = 0\) (constant function).
2. \(\frac{d^2 x}{dy^2} p^2 + \frac{dp}{dy} = 0\).

### Step 9: Solve the second case
From the second case, we can solve for \(\frac{dp}{dy}\):

\[ \frac{dp}{dy} = -\frac{d^2 x}{dy^2} p^2. \]

### Conclusion
Thus, we conclude that:

\[ \frac{d^2 x}{dy^2} \left( \frac{dy}{dx} \right)^3 + \frac{d^2 y}{dx^2} = 0. \]

This means the original equation is satisfied, and we have shown that the expression equals zero.

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