The singular solution of $(x^{2}-1)p^{2}-2xyp+y^{2}-1=0$ is

Question

Accepted Answer

To solve the problem of finding the singular solution of the given differential equation $(x^{2}-1)p^{2}-2xyp+y^{2}-1=0$, where $p = \frac{dy}{dx}$, we need to determine the conditions under which the equation has a singular solution.

### Explanation for Option 1: $x^{2}+y^{2}=1$

1. **Singular Solution Concept**: A singular solution is a solution that cannot be obtained by the general solution of the differential equation. It often arises when the discriminant of the quadratic in $p$ becomes zero, leading to a repeated root.

2. **Analyzing the Given Equation**: The equation $(x^{2}-1)p^{2}-2xyp+y^{2}-1=0$ is quadratic in $p$. For a singular solution, the discriminant of this quadratic equation must be zero.

3. **Discriminant Calculation**: The discriminant $\Delta$ of the quadratic equation $Ap^2 + Bp + C = 0$ is given by $B^2 - 4AC$. Here, $A = x^2 - 1$, $B = -2xy$, and $C = y^2 - 1$.

$$
   \Delta = (-2xy)^2 - 4(x^2 - 1)(y^2 - 1)
   $$
   $$
   = 4x^2y^2 - 4(x^2y^2 - x^2 - y^2 + 1)
   $$
   $$
   = 4x^2y^2 - 4x^2y^2 + 4x^2 + 4y^2 - 4
   $$
   $$
   = 4(x^2 + y^2 - 1)
   $$

For the discriminant to be zero, we require:
   $$
   4(x^2 + y^2 - 1) = 0 \implies x^2 + y^2 = 1
   $$

4. **Conclusion**: The condition $x^2 + y^2 = 1$ ensures that the discriminant is zero, indicating a singular solution. Therefore, Option 1 is correct.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $x^{2}+y^{2}=4$**

If $x^2 + y^2 = 4$, then substituting into the discriminant condition gives:
  $$
  4(x^2 + y^2 - 1) = 4(4 - 1) = 12 \neq 0
  $$
  This does not satisfy the condition for a singular solution.

- **Option 3: $x^{2}-y^{2}=1$**

Substituting $x^2 - y^2 = 1$ into the discriminant condition:
  $$
  4(x^2 + y^2 - 1) \neq 0
  $$
  This does not lead to a zero discriminant, hence not a singular solution.

- **Option 4: $x^{2}-y^{2}=4$**

Similarly, substituting $x^2 - y^2 = 4$ gives:
  $$
  4(x^2 + y^2 - 1) \neq 0
  $$
  This also does not satisfy the singular solution condition.

In conclusion, only Option 1, $x^2 + y^2 = 1$, satisfies the condition for the singular solution of the given differential equation, as it results in a zero discriminant.