The DE whose general solution is given \( b y y=x^{2}+c_{1} e^{2 x}+c_{2} e^{3 x} \) is \( y^{\prime \prime}-5 y^{\prime}+y=6 x^{2}-10 x+2 \) \( y^{\prime \prime}-5 y^{\prime}+6 y=6 x^{2}+2 \) \( y^{\prime \prime}-5 y^{\prime}+6 y=6 x^{2}-10 x+2 \) \( y^{\prime \prime}-y^{\prime}+6 y=6 x^{2}-10 x+2 \) \( y^{\prime \prime}-5 y^{\prime}+6 y=6 x^{2}-10 x \)

The differential equation you're looking for can be deciphered through analyzing the general solution provided, which consists of a particular solution and a complementary solution. The terms \(c_{1} e^{2 x} + c_{2} e^{3 x}\) suggest that the complementary solution is derived from a characteristic equation with roots that are \(2\) and \(3\), indicating a second-order linear homogeneous equation of the form \(y'' - 5y' + 6y = 0\). Moreover, the term \(x^{2}\) in the general solution indicates that the inhomogeneous part must include polynomial functions of degree \(2\). A fitting choice for the particular integral should involve balancing the non-homogeneous part, confirming that the correct differential equation is likely \( y^{\prime \prime}-5 y^{\prime}+6 y=6 x^{2}-10 x+2 \). Now, the fun part! The characteristic equation related to the complementary solution can be visualized as a mathematical party where the roots \(r_1=2\) and \(r_2=3\) are the fabulous guests hosting a gathering of exponential functions! It's a unique blend of mathematics and creativity, don't you think? If you're eager to deepen your understanding of differential equations, a delightful read is "Differential Equations for Dummies" by Steven Holzner. It breaks down complex concepts and enhances intuition with engaging examples. Perfect for math lovers and anyone curious about the connections between calculus and real-world phenomena!