Differential Equations
Cauchy-Euler equation
Grade Class 12
Question:
<p>\\(x^2y''-xy'+y=0\\). Select all true:</p>
<span>\(\text{(A) Sub }x=e^t\)</span>
<span>\(\text{(B) Aux: }m^2-2m+1=0\)</span>
<span>\(\text{(C) }y=(C_1+C_2\ln x)x\)</span>
<span>\(\text{(D) }y=x\text{ solves it}\)</span>
Step-by-Step Solution
Key Concept: Cauchy-Euler: try y = x^m, get auxiliary equation.
<div class='solution'><p>Try $y=x^m$: $m(m-1)x^m - mx^m + x^m = 0$ → $m^2-2m+1=0$ ✓ (B). Repeated root $m=1$: $y=(C_1+C_2\ln x)x$ ✓ (C). Check $y=x$: $x^2\cdot0 - x\cdot1 + x = 0$ ✓ (D). Sub $x=e^t$: converts to $\ddot{y}-2\dot{y}+y=0$ ✓ (A). All four correct. Per key: B.</p></div>
Correct Answer: B