Putnam 1957 B6

The curve $y=y(x)$ satisfies $y'(0)=1.$ It satisfies the differential equation $(x^2 +9)y'' +(x^2 +4)y=0.$ Show that it crosses the $x$ -axis between $$x= \frac{3}{2} \pi \;\; \text{and} \;\; x= \sqrt{\frac{63}{53}} \pi.$$