91Ó°ÊÓ

In each exercise, obtain solutions valid for \(x>0\). $$2 x y^{\prime \prime}+(1+2 x) y^{\prime}-3 y=0$$

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

Obtain two linearly independent solutions valid for \(x>0\) unless otherwise instructed. $$ x^{2} y^{\prime \prime}+x(x-1) y^{\prime}+(1-x) y=0 $$.

Obtain two linearly independent solutions valid near the origin for \(x>0\). Always state the region of validity of each solution that you obtain. $$ 8 x^{2} y^{\prime \prime}+10 x y^{\prime}-(1+x) y=0 $$.

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.") \(x^{2}(x-2) y^{\prime \prime}+3(x-2) y^{\prime}+y=0\)

Obtain the general solution near \(x=0\) except when instructed otherwise. State the region of validity of each solution. $$x(1-x) y^{\prime \prime}-3 y^{\prime}+2 y=0$$

In each exercise, obtain solutions valid for \(x>0\). $$y^{\prime \prime}+2 x y^{\prime}-8 y=0$$

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.") \(x^{2}(x-4)^{2} y^{\prime \prime}+3 x y^{\prime}-(x-4) y=0\)

Obtain two linearly independent solutions valid for \(x>0\) unless otherwise instructed. $$ x(x-2) y^{\prime \prime}+2(x-1) y^{\prime}-2 y=0 $$.

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