**Educational Exercise: Differential Equations** **Objective: Solve the following differential equations.** 1. \( x^2 y'' + xy' + (9x^2 - 4)y = 0 \) 2. \( x^2 y'' + xy' + (36x^2 - \frac{1}{4})y = 0 \) 3. **Hint:** For equation \( x^2 y'' + xy' + (\alpha^2 x^2 - \nu^2)y = 0 \) Solve using the general form: \[ y'' + \frac{1-2\alpha}{x} y' + \left[ \frac{b^2}{c^2 x^{2}} (\beta^2 - 2) + \frac{a^2 - p^2 c^2}{x^2} \right] y = 0 \] 4. \( xy'' + 2y' + 4y = 0 \) 5. \( xy'' - xy' + xy = 0 \) 6. \( 4x^2 y'' + (16x^2 + 1)y = 0 \) **Explanation of Symbols:** - \( y'' \) and \( y' \) denote the second and first derivatives of \( y \) with respect to \( x \), respectively. - The constants \(\alpha\), \(\nu\), \(a\), \(b\), \(c\), \(p\), and \(\beta\) represent known parameters or variables specific to each problem. - The equations are to be solved for the function \( y(x) \). **Approach:** - Identify suitable methods for solving second-order linear differential equations, which might involve recognizing standard forms, applying the method of undetermined coefficients, or using series solutions for specific parameter values. **Note:** The equations provided are classic forms found in mathematical physics and engineering. The solutions often involve special functions depending on the parameters used.