91Ó°ÊÓ

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

Use the power series method to find solutions valid for \(x > 0\). What is causing the recurrence relations to degenerate into one-term relations? $$ 9 x^{2} y^{\prime \prime}+2 y=0. $$

Use the power series method to find solutions valid for \(x > 0\). What is causing the recurrence relations to degenerate into one-term relations? $$ 2 x^{2} y^{\prime \prime}+5 x y^{\prime}-2 y=0. $$

In each exercise, obtain solutions valid for \(x > 0\). $$ 2 x^{2} y^{\prime \prime}-x(2 x+7) y^{\prime}+2(x+5) y=0. $$

Obtain \(d y / d x\) and \(d^{2} y / d x^{2}\) in terms of derivatives of \(y\) with respect to a new independent variable \(t\) related to \(x\) by \(t=\ln x\) for \(x > 0\).

In each exercise, obtain solutions valid for \(x > 0\). $$ \left(1-x^{2}\right) y^{\prime \prime}-10 x y^{\prime}-18 y=0. $$

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

In each exercise, obtain solutions valid for \(x > 0\). $$ 2 x(1-x) y^{\prime \prime}+(1-2 x) y^{\prime}+8 y=0. $$

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

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