91Ó°ÊÓ

Find two linearly independent solutions, valid for \(x>0,\) unless otherwise instructed. $$ 4 x^{2} y^{\prime \prime}+2 x(2-x) y^{\prime}-(1+3 x) y=0 $$

Obtain two linearly independent solutions valid near the origin for \(x>0\). Always state the region of validity of each solution that you obtain. $$ 4 x y^{\prime \prime}+3 y^{\prime}+3 y=0 $$.

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

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

Solve each equation for \(x>0\) unless otherwise instructed. \(x^{2} y^{\prime \prime}+x(1+x) y^{\prime}-\left(1-3 x+6 x^{2}\right) y=0\).

Obtain two linearly independent solutions valid for \(x>0\) unless otherwise instructed. $$ x^{2} y^{\prime \prime}+3 x y^{\prime}+\left(1+4 x^{2}\right) y=0 $$.

The singular points in the finite plane have already been located and classified. For each equation, determine whether the point at infinity is an ordinary point (O.P.), a regular singular point point (R.S.P.), or an irregular singular point (I.S.P.). Do not solve the problems. \(x^{2} y^{\prime \prime}+y=0\). (Exercise 4, Section 18.1.)

Obtain two linearly independent solutions valid for \(x>0\) unless otherwise instructed. $$ x(1+x) y^{\prime \prime}+(1+5 x) y^{\prime}+3 y=0 $$.

Obtain two linearly independent solutions valid near the origin for \(x>0\). Always state the region of validity of each solution that you obtain. $$ 2 x^{2}(1-x) y^{\prime \prime}-x(1+7 x) y^{\prime}+y=0 $$.

Obtain the general solution near \(x=0\) except when instructed otherwise. State the region of validity of each solution. $$x(1+x) y^{\prime \prime}+(x+5) y^{\prime}-4 y=0$$