Hello everyone!
I'm trying to solve \(\displaystyle y"+x^{2} y=0\), given \(\displaystyle y=u\sqrt{x}\) and \(\displaystyle \frac{x^{2}}{2}=z\). When I replace y with \(\displaystyle u\sqrt{x}\) I get \(\displaystyle u"+\frac{u'}{x} +u (x^{2}-\frac{1}{4x^{2}})\). The only difference to a Bessel function would be the term multiplying u: \(\displaystyle x^{2}-\frac{1}{(4x^{2})}\). If I replace just that one by z, I get \(\displaystyle (2z-\frac{1}{8z})\). If I factor 2 out I get \(\displaystyle 2(z-\frac{1}{16z})\). This still doesn't look like Bessel's equation to me because it's not in the form \(\displaystyle 1-\frac{n^{2}}{z}\) (thinking that z is a function of \(\displaystyle x^{2}\)). Is it Bessel's equation?
Further, \(\displaystyle u(x)=C_{1} J_{n}(x)+C_{2}J_{n}(x)\) and then \(\displaystyle y(x)=\sqrt{x}[C_{1}J_{n}(z)+C_{2}J_{n}(z)]=\sqrt{x}[C_{1}J_{n}(\frac{x^{2}}{2})+C_{2}J_{n}(\frac{x^{2}}{2})]\). Why is n=1/4?
Thank you in advance for looking and for your help.
I'm trying to solve \(\displaystyle y"+x^{2} y=0\), given \(\displaystyle y=u\sqrt{x}\) and \(\displaystyle \frac{x^{2}}{2}=z\). When I replace y with \(\displaystyle u\sqrt{x}\) I get \(\displaystyle u"+\frac{u'}{x} +u (x^{2}-\frac{1}{4x^{2}})\). The only difference to a Bessel function would be the term multiplying u: \(\displaystyle x^{2}-\frac{1}{(4x^{2})}\). If I replace just that one by z, I get \(\displaystyle (2z-\frac{1}{8z})\). If I factor 2 out I get \(\displaystyle 2(z-\frac{1}{16z})\). This still doesn't look like Bessel's equation to me because it's not in the form \(\displaystyle 1-\frac{n^{2}}{z}\) (thinking that z is a function of \(\displaystyle x^{2}\)). Is it Bessel's equation?
Further, \(\displaystyle u(x)=C_{1} J_{n}(x)+C_{2}J_{n}(x)\) and then \(\displaystyle y(x)=\sqrt{x}[C_{1}J_{n}(z)+C_{2}J_{n}(z)]=\sqrt{x}[C_{1}J_{n}(\frac{x^{2}}{2})+C_{2}J_{n}(\frac{x^{2}}{2})]\). Why is n=1/4?
Thank you in advance for looking and for your help.