Step 1: Understanding the Concept:
The given differential equation is not in a standard form like linear, separable, or exact. It is a type of equation that is solvable for y. We can try to rearrange it and then differentiate with respect to x to solve it.
Step 2: Key Formula or Approach:
1. Rearrange the equation to isolate \(p\).
2. This will result in a homogeneous differential equation.
3. Solve the homogeneous equation using the substitution \(y=vx\).
Step 3: Detailed Explanation:
The given equation is:
\[ y = x(p + \sqrt{1+p^2}) \]
Divide by \(x\):
\[ \frac{y}{x} = p + \sqrt{1+p^2} \]
\[ \frac{y}{x} - p = \sqrt{1+p^2} \]
Square both sides:
\[ \left(\frac{y}{x} - p\right)^2 = 1+p^2 \]
\[ \frac{y^2}{x^2} - 2\frac{y}{x}p + p^2 = 1+p^2 \]
Cancel \(p^2\) from both sides:
\[ \frac{y^2}{x^2} - 2\frac{y}{x}p = 1 \]
Substitute \(p = \frac{dy}{dx}\):
\[ \frac{y^2}{x^2} - \frac{2y}{x}\frac{dy}{dx} = 1 \]
Multiply by \(x^2\):
\[ y^2 - 2xy\frac{dy}{dx} = x^2 \]
Rearrange to solve for \(\frac{dy}{dx}\):
\[ 2xy\frac{dy}{dx} = y^2 - x^2 \]
\[ \frac{dy}{dx} = \frac{y^2 - x^2}{2xy} \]
This is a homogeneous differential equation. Let \(y = vx\). Then \(\frac{dy}{dx} = v + x\frac{dv}{dx}\).
Substitute these into the equation:
\[ v + x\frac{dv}{dx} = \frac{(vx)^2 - x^2}{2x(vx)} = \frac{v^2x^2 - x^2}{2vx^2} = \frac{x^2(v^2 - 1)}{2vx^2} = \frac{v^2-1}{2v} \]
\[ x\frac{dv}{dx} = \frac{v^2-1}{2v} - v = \frac{v^2 - 1 - 2v^2}{2v} = \frac{-v^2 - 1}{2v} = -\frac{v^2+1}{2v} \]
Separate the variables:
\[ \frac{2v}{v^2+1}dv = -\frac{dx}{x} \]
Integrate both sides:
\[ \int \frac{2v}{v^2+1}dv = -\int \frac{dx}{x} \]
The left integral is of the form \(\int \frac{f'(v)}{f(v)}dv = \ln|f(v)|\).
\[ \ln(v^2+1) = -\ln|x| + \ln|C| \]
\[ \ln(v^2+1) = \ln\left|\frac{C}{x}\right| \]
\[ v^2+1 = \frac{C}{x} \]
Substitute back \(v = y/x\):
\[ \left(\frac{y}{x}\right)^2 + 1 = \frac{C}{x} \]
\[ \frac{y^2}{x^2} + 1 = \frac{C}{x} \]
Multiply by \(x^2\):
\[ y^2 + x^2 = Cx \implies x^2 + y^2 - Cx = 0 \]
Note on Discrepancy: The derived solution is a family of circles passing through the origin. However, none of the options match this result. The provided answer key marks option (D) as correct. Let's check if there's an error in the problem statement or the options provided. The steps taken to convert the original equation into the homogeneous form and its subsequent solution are standard and correct. There seems to be a significant inconsistency between the question and the provided options/answer. It is likely that the question or the options are incorrect.
Step 4: Final Answer:
The rigorous solution to the differential equation is \(x^2 + y^2 - Cx = 0\). As this is not an option, and based on the provided answer key, there is an error in the question paper. We select option (D) as per the key, but note the mathematical derivation leads to a different result.