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Magazine Problem 20
Find two linearly independent Frobenius series solutions (for \(x>0\) ) of each of the differential equations in Problems 17 through \(26 .\) $$ 3 x v^{\prime \prime}+2 v^{\prime}+2 v=0 $$
Problem 20
First derive a recurrence relation giving \(c n\) for \(n \geqq 2\) in terms of \(c_{0}\) or \(c_{1}\) (or both). Then apply the given initial conditions to find the values of \(c_{0}\) and \(c_{1}\). Next determine \(c_{n}\) and, finally, identify the particular solution in terms of familiar elementary functions. $$ y^{\prime \prime}-4 y=0 ; y(0)=2, y^{\prime}(0)=0 $$
Problem 21
Find two linearly independent Frobenius series solutions (for \(x>0\) ) of each of the differential equations in Problems 17 through \(26 .\) $$ 2 x^{2} y^{\prime \prime}+x y^{\prime}-\left(1+2 x^{2}\right) y=0 $$
Problem 22
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.
Problem 22
Find two linearly independent Frobenius series solutions (for \(x>0\) ) of each of the differential equations in Problems 17 through \(26 .\) $$ 2 x^{2} y^{\prime \prime}+x y^{\prime}-\left(3-2 x^{2}\right) y=0 $$
Problem 23
Find two linearly independent Frobenius series solutions (for \(x>0\) ) of each of the differential equations in Problems 17 through \(26 .\) $$ 6 x^{2} y^{\prime \prime}+7 x y^{\prime}-\left(x^{2}+2\right) y=0 $$
Problem 23
Show that the equation $$ x^{2} y^{\prime \prime}+x^{2} y^{\prime}+y=0 $$ has no power series solution of the form \(y=\sum c_{n} x^{n}\).
Problem 23
Prove that $$ J_{1}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (\theta-x \sin \theta) d \theta $$ by showing that the right-hand side satisfies Bessel's equation of order 1 and that its derivative has the value \(J_{1}^{\prime}(0)\) when \(x=0\). Explain why this constitutes a proof.
Problem 23
Find a three-term recurrence relation for solutions of the form \(y=\sum c_{n} x^{n} .\) Then find the first three nonzero terms in each of two linearly independent solutions. $$ y^{\prime \prime}+(1+x) y=0 $$
Problem 24
Find a three-term recurrence relation for solutions of the form \(y=\sum c_{n} x^{n} .\) Then find the first three nonzero terms in each of two linearly independent solutions. $$ \left(x^{2}-1\right) y^{\prime \prime}+2 x y^{\prime}+2 x y=0 $$
