91Ó°ÊÓ

In each exercise, obtain solutions valid for \(x>0\). $$x\left(1-x^{2}\right) y^{\prime \prime}-\left(7+x^{2}\right) y^{\prime}+4 x y=0$$

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

In each exercise, obtain solutions valid for \(x>0\). $$2 x^{2} y^{\prime \prime}-x(1+2 x) y^{\prime}+(1+4 x) y=0$$

Obtain two linearly independent solutions valid near the origin for \(x>0\). Always state the region of validity of each solution that you obtain. $$ 2 x y^{\prime \prime}+\left(1-2 x^{2}\right) y^{\prime}-4 x y=0 $$.

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

Find two linearly independent solutions, valid for \(x>0,\) unless otherwise instructed. \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0 .\) This is Bessel's equation of index one. (See also Sections 19.5 and \(19.6 .)\)

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

Solve about \(x=4: 4(x-4)^{2} y^{\prime \prime}+(x-4)(x-8) y^{\prime}+x y=0 .\)

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

Find two linearly independent solutions, valid for \(x>0,\) unless otherwise instructed. \(x^{2} y^{\prime \prime}-5 x y^{\prime}+(8+5 x) y=0\)