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(Steps Shown) The Bessel function of order 1 is defined by J_1(x)=∑_n=0^∞ ((-1)^n x^2 n+1)/(n !(n+1) ! 2^2 n+1) Show that J_1 satisfies the differential

Question: The Bessel function of order 1 is defined by

\[J_{1}(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{n !(n+1) ! 2^{2 n+1}}\]
  1. Show that \(J_{1}\) satisfies the differential equation
    \[x^{2} J_{1}^{\prime \prime}(x)+x J_{1}^{\prime}(x)+\left(x^{2}-1\right) J_{1}(x)=0\]
  2. Show that \(J_{0}^{\prime}(x)=-J_{1}(x)\).

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