Answer: x m (power function).
- A e<sup>mx</sup>, the solution form for constant-coefficient equations
- B x<sup>m</sup> (power function)
- C sinx, a trigonometric solution form
- D xe<sup>x</sup>, a solution form for repeated roots
Correct answer: B. x<sup>m</sup> (power function)
Explanation: For Euler-Cauchy equation x²y''+bxy'+cy=0, substitute y=xᵐ. Then y'=mxᵐ⁻¹, y''=m(m−1)xᵐ⁻². Equation reduces to m(m−1)+bm+c=0 (indicial equation). Roots m determine the power-function solutions xᵐ.
The general solution y=Ax² represents an entire FAMILY of curves, one for each value of the arbitrary constant A; a particular solution (fixed by an initial condition) selects exactly one curve from this family.
Concept context
Equations involving derivatives and their solutions