91Ó°ÊÓ

Find the general solution except when the exercise stipulates otherwise. $$ \left(2 D^{3}-D^{2}+36 D-18\right) y=0. $$

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

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(D^{4}-2 D^{3}-13 D^{2}+38 D-24\right) y=0. $$

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(6 D^{4}+23 D^{3}+28 D^{2}+13 D+2\right) y=0. $$

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

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

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

Find the general solution except when the exercise stipulates otherwise. $$ \left(D^{3}+D^{2}+4 D+4\right) y=0 ; \text { when } x=0, y=0, y^{\prime}=-1, y^{\prime \prime}=5. $$

Find the general solution. When the operator \(D\) is used, it is implied that the independent variable is \(x\). $$ \left(4 D^{4}-45 D^{2}-70 D-24\right) y=0. $$

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