91Ó°ÊÓ

For each equation, list all the singular points in the finite plane. $$ 4 y^{\prime \prime}+y=0 $$.

Find the general solution valid near the origin. Always state the region of validity of the solution. $$ \left(1+2 x^{2}\right) y^{\prime \prime}-5 x y^{\prime}+3 y=0 $$

Find the general solution valid near the origin. Always state the region of validity of the solution. $$ y^{\prime \prime}+x^{2} y=0 $$

For each equation, list all the singular points in the finite plane. $$ x^{2}\left(x^{2}-9\right) y^{\prime \prime}+3 x y^{\prime}-y=0 $$.

For each equation, list all the singular points in the finite plane. $$ x^{2}\left(1+4 x^{2}\right) y^{\prime \prime}-4 x y^{\prime}+y=0 $$.

Find the general solution valid near the origin. Always state the region of validity of the solution. $$ \left(1-4 x^{2}\right) y^{\prime \prime}+6 x y^{\prime}-4 y=0 $$

For each equation, list all the singular points in the finite plane. $$ (2 x+1)(x-3) y^{\prime \prime}-y^{\prime}+(2 x+1) y=0 $$.

Find the general solution valid near the origin. Always state the region of validity of the solution. $$ y^{\prime \prime \prime}+x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0 $$

For each equation, list all the singular points in the finite plane. $$ x^{3}\left(x^{2}-4\right)^{2} y^{\prime \prime}+2\left(x^{2}-4\right) y^{\prime}-x y=0 $$.

For each equation, list all the singular points in the finite plane. $$ x\left(x^{2}+1\right)^{2} y^{\prime \prime}-y=0 $$.