Differential Equations
Singular solution
Grade Class 12
Question:
<p>Singular solution of \\(\\left(\\dfrac{dy}{dx}\\right)^2 - x\\dfrac{dy}{dx} + y = 0\\).</p>
<span>\(y = x^2/4\)</span>
<span>\(y = x^2\)</span>
<span>\(y = x^2/2\)</span>
<span>\(y = 2x^2\)</span>
Step-by-Step Solution
Key Concept: p-discriminant method: eliminate p from ODE and dp/dx = 0.
<div class='solution'><p>ODE: \(p^2-xp+y=0\) where \(p=dy/dx\). Singular solution: discriminant = 0. From \(p^2-xp+y=0\): treat as quadratic in \(p\): discriminant \(x^2-4y=0\) → \(y=x^2/4\). <strong>Verify:</strong> \(y=x^2/4\) → \(y'=x/2\). Sub: \((x/2)^2-x(x/2)+x^2/4=x^2/4-x^2/2+x^2/4=0\) ✓. <strong>Answer: (1)</strong> \(y=x^2/4\).</p><p class='key-concept'>🔑 Key Concept: For \(p^2+Pp+Q=0\), the singular solution is \(P^2-4Q=0\) (p-discriminant).</p></div>
Correct Answer: 1