91Ó°ÊÓ

Obtain two linearly independent solutions valid near the origin for \(x>0\). Always state the region of validity of each solution that you obtain. $$ x(4-x) y^{\prime \prime}+(2-x) y^{\prime}+4 y=0 $$.

In each exercise, obtain solutions valid for \(x>0\). $$4 x^{2} y^{\prime \prime}-2 x(2+x) y^{\prime}+(3+x) y=0$$

For each equation, locate and classify all its singular points in the finite plane. (See Section 18.10 for the concept of a singular point "at infinity.") \(x^{3} y^{\prime \prime}+4 y=0\)

Find solutions valid for large positive \(x\) unless otherwise instructed. $$ 2 x^{2}(x-1) y^{\prime \prime}+x(5 x-3) y^{\prime}+(x+1) y=0 $$.

Obtain the general solution near \(x=0\) except when instructed otherwise. State the region of validity of each solution. $$x y^{\prime \prime}+(3+2 x) y^{\prime}+4 y=0$$

Find two linearly independent solutions, valid for \(x>0,\) unless otherwise instructed. \(x y^{\prime \prime}+(3-x) y^{\prime}-5 y=0\)

Obtain two linearly independent solutions valid near the origin for \(x>0\). Always state the region of validity of each solution that you obtain. $$ 3 x^{2} y^{\prime \prime}+x y^{\prime}-(1+x) y=0 $$.

In each exercise, obtain solutions valid for \(x>0\). $$x^{2} y^{\prime \prime}-x\left(1+x^{2}\right) y^{\prime}+\left(1-x^{2}\right) y=0$$

In each exercise, obtain solutions valid for \(x>0\). $$2 x y^{\prime \prime}+y^{\prime}+y=0$$

Find two linearly independent solutions, valid for \(x>0,\) unless otherwise instructed. \(9 x^{2} y^{\prime \prime}-15 x y^{\prime}+7(1+x) y=0\)