Skip to content
Part CCSIR NET December 2025the-indicial-roots-3-over-2-and-minus-1-pin-down-which-solution-is-regular-and-which-blows-up-at-the-origin

The indicial roots 3 over 2 and minus 1 pin down which solution is regular and which blows up at the origin

Suppose y1(x)y_{1}(x) and y2(x)y_{2}(x) are two linearly independent solutions of the differential equation x2y′′+[(1+sin⁡x)x/2]y′−(3/2)(cos⁡x)y=0,x>0x^{2}y'' + [(1+\sin x)x/2]y' - (3/2)(\cos x)y = 0, x>0, satisfying y2(0)=0y_{2}(0)=0. Then which of the following statements are true?

  1. A.lim⁡(x→0+)y2(x)/(x2y1(x))\lim(x\to0+) y_{2}(x)/(x^{2}y_{1}(x)) exists.✓
  2. B.lim⁡(x→0+)y2(x)/(\lim(x\to0+) y_{2}(x)/(xy1(x))_{1}(x)) exists.✓
  3. C.lim(x→0+) xy1(x)/y2(x)_{1}(x)/y_{2}(x) does NOT exist.✓
  4. D.lim⁡(x→0+)x2y1(x)/y2(x)\lim(x\to0+) x^{2}y_{1}(x)/y_{2}(x) exists.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: W(f, g) ≡ 0 implies f and g are linearly dependent

More on this topic

The chapter behind this: Linear ODE, Wronskian and systems — free to read

From Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

ShareWhatsAppTelegram