Questions: Sabiendo que la función y₁(x)=x sen(ln (x)) es solución de la ecuación diferencial: x² y''-x y'+2 y=0, x>0 Determine su solución general aplicando fórmula de Abel vista en el curso.

To determine the general solution of the given differential equation using Abel's formula, we need to follow these steps:

1. Identify the given particular solution \( y_1(x) = x \sin(\ln(x)) \).
2. Use Abel's formula to find the second linearly independent solution \( y_2(x) \).
3. Construct the general solution as a linear combination of \( y_1(x) \) and \( y_2(x) \).

La ecuación diferencial dada es: \[ x^{2} y^{\prime \prime}-x y^{\prime}+2 y=0, \quad x>0 \] Y se nos da una solución particular: \[ y_{1}(x)=x \operatorname{sen}(\ln (x)) \]

La fórmula de Abel nos permite encontrar una segunda solución linealmente independiente \( y_2(x) \) de una ecuación diferencial de segundo orden cuando ya conocemos una solución \( y_1(x) \).

La fórmula de Abel para encontrar \( y_2(x) \) es: \[ y_2(x) = y_1(x) \int \frac{e^{-\int P(x) \, dx}}{(y_1(x))^2} \, dx \] donde la ecuación diferencial está en la forma estándar: \[ y'' + P(x)y' + Q(x)y = 0 \]

Primero, reescribimos la ecuación diferencial en la forma estándar: \[ x^{2} y^{\prime \prime}-x y^{\prime}+2 y=0 \] Dividimos todo por \( x^2 \): \[ y^{\prime \prime} - \frac{1}{x} y^{\prime} + \frac{2}{x^2} y = 0 \] Aquí, \( P(x) = -\frac{1}{x} \) y \( Q(x) = \frac{2}{x^2} \).

Calculamos \( e^{-\int P(x) \, dx} \): \[ P(x) = -\frac{1}{x} \] \[ \int P(x) \, dx = \int -\frac{1}{x} \, dx = -\ln|x| = -\ln(x) \quad (\text{para } x > 0) \] \[ e^{-\int P(x) \, dx} = e^{-\left(-\ln(x)\right)} = e^{\ln(x)} = x \]

Usamos la fórmula de Abel: \[ y_2(x) = y_1(x) \int \frac{x}{(y_1(x))^2} \, dx \] Sabemos que \( y_1(x) = x \operatorname{sen}(\ln(x)) \), entonces: \[ (y_1(x))^2 = \left(x \operatorname{sen}(\ln(x))\right)^2 = x^2 \operatorname{sen}^2(\ln(x)) \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{x}{x^2 \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^2(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx \] \[ y_2(x) = x \operatorname{sen}(\ln(x)) \int \frac{1}{x \operatorname{sen}^(\ln(x))} \, dx = x \operatorname{sen}(\ln(x)) \int \frac{1