$(a)y^{\text{'}\text{'}}+{}^{2}xy=0,x>0$;$t^2{w}^{\text{'}\text{'}}+t{w}^{\text{'}}+(t^2-\frac{1}{9})w=0,t>0$

Differentiating $y=x^{\frac{1}{2}}w(\frac{2}{3}{x}^{\frac{3}{2}})$ with respect $to\frac{2}{3}{x}^{\frac{3}{2}}$ then

$\{$:$y^{\text{'}}=(,)w^{\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})+1,)w(\frac{2}{3}{x}^{\frac{3}{2}})$

and

$y^{\text{'}\text{'}}=(,)w^{\text{'}\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})+(,)w^{\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})+(,)w(\frac{2}{3}{x}^{\frac{3}{2}}).$

Then, after simplifying $we$ have

$\{$:$y^{\text{'}\text{'}}+{}^{2}xy=[x^{\frac{3}{2}}{w}^{\text{'}\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})+\frac{3}{2}{w}^{\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})+1,)w(\frac{2}{3}{x}^{\frac{3}{2}})]=0.$

Letting $t=\frac{2}{3}{x}^{\frac{3}{2}}$ this differential equation becomes $\frac{3}{2}\frac{}{t}[t^2{w}^{\text{'}\text{'}}(t)+t{w}^{\text{'}}(t)+(t^2-\frac{1}{9})w(t)]=0,t>0.$

$(b)y^{\text{'}\text{'}}-{}^{2}xy=0,x>0$;$t^2{w}^{\text{'}\text{'}}+t{w}^{\text{'}}-(t^2+\frac{1}{9})w=0,t>0$

Using the same substitution $y=x^{\frac{1}{2}}w(\frac{2}{3}{x}^{\frac{3}{2}})$ then after simplifying $we$ have the following differential equation.

$\{$:$y^{\text{'}\text{'}}-{}^{2}xy=[x^{\frac{3}{2}}{w}^{\text{'}\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})+\frac{3}{2}{w}^{\text{'}}(\frac{2}{3}{x}^{\frac{3}{2}})-1,)w(\frac{2}{3}{x}^{\frac{3}{2}})]=0$

Letting $t=\frac{2}{3}{x}^{\frac{3}{2}}$ this differential equation becomes $\frac{3}{2}\frac{}{t}[t^2{w}^{\text{'}\text{'}}(t)+t{w}^{\text{'}}(t)-(t^2+\frac{1}{9})w(t)]=0,t>0.$