Example2: \( y^{\prime \prime}+y^{\prime}-2 y=x e^{x} \)

To solve the differential equation \( y^{\prime \prime}+y^{\prime}-2 y=x e^{x} \), we start by finding the complementary solution (homogeneous part) by solving \( y^{\prime \prime} + y^{\prime} - 2y = 0 \). The characteristic equation is \( r^2 + r - 2 = 0 \), which factors into \( (r-1)(r+2) = 0 \), yielding roots \( r = 1 \) and \( r = -2 \). Thus, the complementary solution is \( y_c = C_1 e^{x} + C_2 e^{-2x} \). Next, for the particular solution \( y_p \) for the non-homogeneous part \( x e^{x} \), we can use the method of undetermined coefficients. Guess a solution of the form \( y_p = (Ax + B)e^{x} \). Upon differentiating and substituting into the original equation, we solve for the coefficients \( A \) and \( B \). By combining the complementary and particular solutions, we find the general solution of the differential equation. Now, let’s get into some exciting facts about it! This equation is a second-order linear ordinary differential equation (ODE), and its solutions can describe phenomena in physics and engineering, like vibrations, waves, or thermal systems. In control theory, similar equations govern the behavior of control systems, allowing engineers to design systems that respond predictively to inputs. Additionally, the method of solving such ODEs with the superposition principle and undetermined coefficients is a common theme in differential equations. If you're interested in diving deeper, look into differential equations textbooks where theories are well-explored, or check online courses that provide problem sets. Places like MIT OpenCourseWare offer fantastic resources!