Which statement about the general real solution is true?

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Multiple Choice

Which statement about the general real solution is true?

Explanation:
For Euler-Cauchy (or equidimensional) equations, try a solution of the form y = x^r. Substituting into the equation gives a quadratic in r. If the roots are complex r = α ± iβ, the real general solution becomes y = x^α [C1 cos(β ln x) + C2 sin(β ln x)]. In this case the roots are r = -1 ± i, so α = -1 and β = 1. That leads to y = x^{-1} [C1 cos(ln x) + C2 sin(ln x)]. This is the correct real general solution because the imaginary part of the roots produces the oscillatory dependence on ln x, scaled by x^{-1}. The other forms would correspond to different root structures (for example, a real exponential in x or using hyperbolic functions) and do not match the given roots.

For Euler-Cauchy (or equidimensional) equations, try a solution of the form y = x^r. Substituting into the equation gives a quadratic in r. If the roots are complex r = α ± iβ, the real general solution becomes y = x^α [C1 cos(β ln x) + C2 sin(β ln x)]. In this case the roots are r = -1 ± i, so α = -1 and β = 1. That leads to y = x^{-1} [C1 cos(ln x) + C2 sin(ln x)]. This is the correct real general solution because the imaginary part of the roots produces the oscillatory dependence on ln x, scaled by x^{-1}. The other forms would correspond to different root structures (for example, a real exponential in x or using hyperbolic functions) and do not match the given roots.

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