91Ó°ÊÓ

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(4 D^{4}-16 D^{3}+7 D^{2}+4 D-2\right) y=0 $$

Find the general solution. \(\left(4 D^{5}-15 D^{3}-5 D^{2}+15 D+9\right) y=0\).

Find the general solution. \(\left(D^{4}-5 D^{2}-6 D-2\right) y=0\).

Find the general solution except when the exercise stipulates otherwise. $$\frac{d^{2} x}{d t^{2}}+k^{2} x=0, k \text { real; when } t=0, x=0, \frac{d x}{d t}=v_{0}$$

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(4 D^{4}+4 D^{3}-13 D^{2}-7 D+6\right) y=0 $$

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(4 D^{5}-8 D^{4}-17 D^{3}+12 D^{2}+9 D\right) y=0 $$

Find the general solution. \(\left(D^{5}-5 D^{4}+7 D^{3}+D^{2}-8 D+4\right) y=0\).

Find the particular solution indicated. \(\left(D^{2}+4 D+4\right) y=0 ;\) when \(x=0, y=1, y^{\prime}=-1\).

Find the general solution except when the exercise stipulates otherwise. $$\frac{d^{2} x}{d t^{2}}+2 b \frac{d x}{d t}+k^{2} x=0, k>b>0 ; \text { when } t=0, x=0, \frac{d x}{d t}=v_{0}$$

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(D^{2}-4 a D+3 a^{2}\right) y=0 ; a \text { real } \neq 0 $$