Chapter 8 - Linear Differential Equations of Order n - 8.1 General Theory for Linear Differential Equations - Problems - Page 503: 14

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We are given $L_1=D+1\\ L_2=D-2x^2$ 1. Determine Ly: $$L_1y = y' + y\\ L_2y=y'-2x^2y$$ 2. Substitute y = $2e^{3x}$: $$L_1(L_2y)=L_1(y'-2x^2y)\\ =y''-4xy-2x^2y'+y'-2x^2y\\ =y''+(1-2x^2)y'-y(4x-2x^2)$$ $$L_2(L_1y)=L_2(y'+y)\\ =y''+y'-2x^(y'+y)\\ =y''+(1-2x^2)y'-2x^2y$$ Then: $$L_1L_2=D^2-D(1-2x^2)-(4x-2x^2)\\ L_2L_1=D^2-D(1-2x^2)-2x^2$$ Hence, $L_1L_2 \ne L_2L_1$