Chapter 6.4, Problem 39E

(a)

To determine

To show: The general solution of the differential equation $x{y}^{″}+\lambda y=0$ on interval $\left(0,\infty \right)$ is $y=c_1\sqrt{x}{J}_1\left(2\sqrt{\lambda }x\right)+c_2\sqrt{x}{Y}_1\left(2\sqrt{\lambda x}\right)$.

(b)

To determine

To verify: The particular solution of the differential equation $x{y}^{″}+\lambda y=0$ in case $\lambda =1$ is $y=\sqrt{x}{J}_1\left(2\sqrt{\lambda }x\right)$.