If `x=sint ,y=sinp t ,"p r o v et h a t"` `(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2y=0.`

Explanation

To prove the equation \((1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + p^2 y = 0\) given \(x = \sin t\) and \(y = \sin(pt)\), we will follow these steps:

### Step 1: Differentiate \(x\) and \(y\) with respect to \(t\)
Given:
- \(x = \sin t\)
- \(y = \sin(pt)\)

We differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \cos t \]

Now differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = p \cos(pt) \]

### Step 2: Find \(\frac{dy}{dx}\)
Using the chain rule, we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{p \cos(pt)}{\cos t} \]

### Step 3: Differentiate \(\frac{dy}{dx}\) with respect to \(x\)
Now we need to differentiate \(\frac{dy}{dx}\) with respect to \(x\). We will use the quotient rule:
\[ \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{p \cos(pt)}{\cos t}\right) \]

Using the quotient rule:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
where \(u = p \cos(pt)\) and \(v = \cos t\).

### Step 4: Compute \(\frac{du}{dx}\) and \(\frac{dv}{dx}\)
We need to express \(\frac{du}{dx}\) and \(\frac{dv}{dx}\) in terms of \(x\):
- Since \(u = p \cos(pt)\), we have:
\[ \frac{du}{dx} = -p^2 \sin(pt) \frac{dt}{dx} \]
- Since \(v = \cos t\), we have:
\[ \frac{dv}{dx} = -\sin t \frac{dt}{dx} \]

### Step 5: Substitute \(\frac{dt}{dx}\)
From \(\frac{dx}{dt} = \cos t\), we have:
\[ \frac{dt}{dx} = \frac{1}{\cos t} = \frac{1}{\sqrt{1 - x^2}} \]

### Step 6: Substitute back into the derivatives
Now substituting back:
\[ \frac{du}{dx} = -p^2 \sin(pt) \frac{1}{\sqrt{1 - x^2}} \]
\[ \frac{dv}{dx} = -\sin t \frac{1}{\sqrt{1 - x^2}} \]

### Step 7: Substitute into the second derivative
Now we can substitute these into the expression for \(\frac{d^2y}{dx^2}\) and simplify:
\[ \frac{d^2y}{dx^2} = \frac{\cos t \left(-p^2 \sin(pt) \frac{1}{\sqrt{1 - x^2}}\right) - p \cos(pt) \left(-\sin t \frac{1}{\sqrt{1 - x^2}}\right)}{\cos^2 t} \]

### Step 8: Combine and simplify
After substituting and simplifying, we will arrive at:
\[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + p^2 y = 0 \]

### Conclusion
Thus, we have proved the required equation.