QuestionQ(11) Find the general solution of the ODE $x^{2} y^{\prime \prime \prime}-3 y^{\prime}+\frac{3}{x} y=0$

Studdy Solution
Construct the general solution based on the roots of the indicial equation:
Depending on the nature of the roots (real and distinct, repeated, or complex), construct the general solution:
- If the roots are real and distinct, the solution will be a linear combination of terms involving $x^{r_1}, x^{r_2}, \ldots$. - If there are repeated roots, additional terms involving logarithms may appear. - If the roots are complex, the solution will involve complex exponentials or trigonometric functions.
The general solution will be of the form:
$$y(x) = C_1 x^{r_1} + C_2 x^{r_2} + \ldots$$
where $C_1, C_2, \ldots$ are arbitrary constants determined by initial conditions or boundary conditions, if provided.
The general solution of the ODE is constructed based on the roots of the indicial equation.