91Ó°ÊÓ

To solve a second-order linear ordinary differential equation, we start by finding the characteristic equation. Assume a solution of the form \( y = e^{rx} \). Substituting this into the differential equation \( (x+1) y^{\prime \prime}+(x-1) y^{\prime}-2 y=0 \) transforms it into an algebraic equation in terms of \( r \). This is our characteristic equation, which can help us find the values of \( r \) that satisfy the differential equation. To break it down further, plug in the assumed solution: \( (x+1) r^2 e^{rx} + (x-1) r e^{rx} - 2 e^{rx} = 0 \). Factor out \( e^{rx} \) (since it can't be zero), which simplifies to: \( r^2 (x+1) + r (x-1) - 2 = 0 \). This is our characteristic equation.

Once we have the characteristic equation, we often get a quadratic equation like \( r^2 + r - 2 = 0 \). To find the values of \( r \) (roots), we can use the quadratic formula: \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula is crucial for solving any quadratic equation. For our characteristic equation, set \( a = 1 \), \( b = 1 \), and \( c = -2 \). Plug these into the quadratic formula: \( r = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm 3}{2} \).We get two roots: \( r_1 = 1 \) and \( r_2 = -2 \). These roots allow us to move to the next step: writing the general solution.

With the roots \( r_1 \) and \( r_2 \) from the quadratic formula, we can write the general solution to our differential equation. For a second-order linear differential equation, the general solution is of the form: \( y = C_1 e^{r_1 x} + C_2 e^{r_2 x} \).Here, \( C_1 \) and \( C_2 \) are constants that can be determined if we have initial conditions or additional information. Substituting our specific roots \( r_1 = 1 \) and \( r_2 = -2 \) into the general form gives us: \( y = C_1 e^{x} + C_2 e^{-2x} \).This is the complete general solution to the original differential equation \( (x+1) y^{\prime \prime}+(x-1) y^{\prime}-2 y=0 \).Understanding how to derive this general solution is critical for solving similar types of differential equations in the future.