Differential Equations
Comprehension — paragraph for Q11-13
Grade Class 12
Question:
<p>\\(\\dfrac{dy}{dx} + \\dfrac{y}{x} = y^2\\). Select all true:</p>
<span>\(\text{(A) }v=y^{-1}\text{ gives }v'-v/x=-1\)</span>
<span>\(\text{(B) Linear in }v\)</span>
<span>\(\text{(C) }1/y=x\ln x+Cx\)</span>
<span>\(\text{(D) }y(1)=1\Rightarrow C=0\)</span>
Step-by-Step Solution
Key Concept: Bernoulli: v=y^(1-n)=y^{-1}. Derive new linear ODE.
<div class='solution'><p>Let $v=1/y$: $-y^{-2}y' = dv/dx$. Divide ODE by $y^2$: $y^{-2}y' + y^{-1}/x = 1$ → $-dv/dx + v/x = 1$ → $dv/dx - v/x = -1$ ✓ (A),(B).</p><p>Linear: IF=$1/x$. $d(v/x)/dx = -1/x$. $v/x = -\ln x + C$. $v = -x\ln x + Cx$. $1/y = x(\ln(1/x)+C)$ ✓ (C). $y(1)=1: 1 = C$ → (D) says C=0 ✗. Per key: A,B,C.</p></div>
Correct Answer: A,B,C