91Ó°ÊÓ

Either apply the method of Example 1 to find two linearly independent Frobenius series solutions, or find one such solution and show (as in Example 2) that a second such solution does not exist. \(x y^{\prime \prime}+(5+3 x) y^{\prime}+3 y=0\)

Find general solutions in powers of \(x\) of the differential equations. State the recurrence relation and the guaranteed radius of convergence in each case. $$ y^{\prime \prime}+x^{2} y=0 $$

Find two linearly independent Frobenius series solutions (for \(x>0\) ) of each of the differential equations in Problems 17 through \(26 .\) $$ 2 x^{2} y^{\prime \prime}+x y^{\prime}-\left(1+2 x^{2}\right) y=0 $$

Prove that $$ J_{0}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (x \sin \theta) d \theta $$ by showing that the right-hand side satisfies Bessel's equation of order zero and has the value \(J_{0}(0)\) when \(x=0\). Explain why this constitutes a proof.

Show that the equation $$ x^{2} y^{\prime \prime}+x^{2} y^{\prime}+y=0 $$ has no power series solution of the form \(y=\sum c_{n} x^{n}\).