91Ó°ÊÓ

A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. When a row operation is applied to a homogeneous system, the new system is still homogeneous.

The given differential equation is ${(D−1)}^2(D+3){(D^2+2D+5)}^2\left[y\right]=0$. To solve this equation we look at its auxillary equation which is ${(m−1)}^2(m+3){(m^2+2m+5)}^2=0$. To get all the solution of its equation we need to solve$m^2+2m+5=0$ .

We have

$m=\frac{−2Â\pm \sqrt{4−20}}{2}=−1Â\pm 2i$

The complete set of solution of auxillary equation is .$\{1,1,−3,−1+2i,−1+2i,−1−2i,−1−2i\}$

To conclude that the general solution of the given differential equation is $y=C_1{e}^x+C_{2}x{e}^x+C_3{e}^{−3x}+C_4{e}^{−x}\sin 2x+C_{5}x{e}^{−x}\sin 2x+C_6{e}^{−x}\cos 2x+C_{7}x{e}^{−x}\cos 2x$, where $C_i(1≤i≤7)$ are arbitrary constants.

The general solution of the given differential equation is$y=C_1{e}^x+C_{2}x{e}^x+C_3{e}^{−3x}+C_4{e}^{−x}\sin 2x+C_{5}x{e}^{−x}\sin 2x+C_6{e}^{−x}\cos 2x+C_{7}x{e}^{−x}\cos 2x$ , where $C_i(1≤i≤7)$ are arbitrary constant.

Hence, the final answer is:$y=C_1{e}^x+C_{2}x{e}^x+C_3{e}^{−3x}+C_4{e}^{−x}\sin 2x+C_{5}x{e}^{−x}\sin 2x+C_6{e}^{−x}\cos 2x+C_{7}x{e}^{−x}\cos 2x$