Question

If $$ y = A t^2 + \frac{B}{t} \quad (A, B \text{ constants}) $$ is a general solution of the differential equation $$ f(t) y'' + g(t) y' + h(t) y = 0, $$ then find the relation between $ g(t), f(t), h(t) $.

• A. \( g(t) - h(t) \)
• B. \( g(t) + f(t) \)
• C. \( g(t) f(t) \)
• D. \( (f(t))^{g(t)} \)

Hint:

Use given solution to find derivatives and substitute back to identify coefficient relations.

Answer 1

The Correct Option is C

Given the solution form \( y = A t^2 + \frac{B}{t} \), find derivatives: \[ y' = 2 A t - \frac{B}{t^2}, \quad y" = 2 A + \frac{2B}{t^3} \] Substitute into the equation: \[ f(t) y" + g(t) y' + h(t) y = 0 \] Collect terms and compare coefficients of \(t^2\) and \(1/t\) to find the relation. The relation turns out to be: \[ g(t) f(t) \]