Without solving the differential equation x²y + xy + (x2−a²)y = 0, determine the linear dependence of its solutions

The given differential equation is $x^2 y'' + x y' + (x^2 - a^2)y = 0$.
This is a second-order linear homogeneous differential equation of the form $P(x)y'' + Q(x)y' + R(x)y = 0$.
Comparing the given equation to the general form, we identify the coefficients:

$$P(x) = x^2$$ $$Q(x) = x$$ $$R(x) = x^2 - a^2$$

Abel's formula states that the Wronskian $W(y_1, y_2)(x)$ of two solutions $y_1(x)$ and $y_2(x)$ to a second-order linear homogeneous differential equation is given by:

$$W(y_1, y_2)(x) = C \exp\left(-\int \frac{Q(x)}{P(x)} dx\right)$$

First, calculate the ratio $\frac{Q(x)}{P(x)}$:

$$\frac{Q(x)}{P(x)} = \frac{x}{x^2} = \frac{1}{x}$$

Next, integrate this ratio:

$$\int \frac{1}{x} dx = \ln|x|$$

Now, substitute this result back into Abel's formula:

$$W(y_1, y_2)(x) = C \exp(-\ln|x|)$$ $$W(y_1, y_2)(x) = C \exp(\ln(|x|^{-1}))$$ $$W(y_1, y_2)(x) = C |x|^{-1}$$ $$W(y_1, y_2)(x) = \frac{C}{|x|}$$

where $C$ is an arbitrary constant determined by the specific solutions $y_1$ and $y_2$.

For two solutions $y_1(x)$ and $y_2(x)$ to be linearly dependent on an interval, their Wronskian must be identically zero on that interval. Conversely, if their Wronskian is non-zero at any point in the interval, they are linearly independent on that interval.
From Abel's formula, we found $W(y_1, y_2)(x) = \frac{C}{|x|}$.

• If $C = 0$, then $W(y_1, y_2)(x) = 0$ for all $x \neq 0$. In this case, the solutions $y_1(x)$ and $y_2(x)$ are linearly dependent. This occurs if one solution is a constant multiple of the other.
• If $C \neq 0$, then $W(y_1, y_2)(x) = \frac{C}{|x|} \neq 0$ for all $x \neq 0$. In this case, the solutions $y_1(x)$ and $y_2(x)$ are linearly independent on any interval that does not contain $x=0$. The point $x=0$ is a singular point for the differential equation because $P(0)=0$. The theory of Wronskians and linear independence/dependence based on it applies on intervals where $P(x) \neq 0$, i.e., on $(-\infty, 0)$ or $(0, \infty)$.

In summary, the solutions to the differential equation $x^2 y'' + x y' + (x^2 - a^2)y = 0$ are linearly independent on any interval not containing $x=0$, provided they are not trivial multiples of each other (i.e., $C \neq 0$). If they are linearly dependent, their Wronskian is identically zero, which implies $C=0$.

The final answer is $\boxed{\text{The solutions are linearly independent on any interval not containing } x=0 \text{ if their Wronskian constant } C \neq 0. \text{ They are linearly dependent if } C=0.}$.