91Ó°ÊÓ

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

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

Find the particular solution indicated. $$ \left(D^{2}-2 D-3\right) y=0 ; \text { when } x=0, y=0, y^{\prime}=-4. $$

Find the particular solution indicated. $$ \left(D^{3}-3 D-2\right) y=0 ; \text { when } x=0, y=0, y^{\prime}=9, y^{\prime \prime}=0. $$

Find the particular solution indicated. $$ \left(D^{4}+3 D^{3}+2 D^{2}\right) y=0 ; \text { when } x=0, y=0, y^{\prime}=4, y^{\prime \prime}=-6, y^{\prime \prime \prime}=14. $$

Find the particular solution indicated. $$ \left(D^{2}-D-6\right) y=0 ; \text { when } x=0, y=0, \text { and when } x=1, y=e^{3}. $$

Find for \(x=1\) the \(y\) value for the particular solution required. $$ \left(D^{2}-2 D-3\right) y=0 ; \text { when } x=0, y=4, y^{\prime}=0. $$

Find for \(x=1\) the \(y\) value for the particular solution required. $$ \left(D^{3}-4 D\right) y=0 ; \text { when } x=0, y=0, y^{\prime}=0, y^{\prime \prime}=2. $$

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

Find for \(x=1\) the \(y\) value for the particular solution required. $$ \left(D^{2}-D-6\right) y=0 ; \text { when } x=0, y=3, y^{\prime}=-1. $$