Answer: xm (power function).
- A emx, the solution form for constant-coefficient equations
- B xm (power function)
- C sinx, a trigonometric solution form
- D xex, a solution form for repeated roots
Correct answer: B. xm (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