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Sage can give us a symbolic solution in terms of Bessel functions:
sage: y=function("y")
sage: var("lambda_")
lambda_
sage: E=diff(y(x),x,2)-x*y(x)==lambda_*y(x)
sage: S=desolve(E,y(x),ivar=x, contrib_ode=True);S
[y(x) == 3/2*(2/3)^(2/3)*(2/3*I*(lambda_ + x)^(3/2))^(1/3)*_K2*sqrt(lambda_ + x)*bessel_I(1/3, 2/3*(lambda_ + x)^(3/2))/((lambda_ + x)^(3/2))^(1/3) + _K1*sqrt(lambda_ + x)*bessel_Y(1/3, 2/3*I*(lambda_ + x)^(3/2))]
sage: var("_K1, _K2")
wich is better seen/understood via \LaTeX:
$$y\left(x\right) = \frac{3 \left(\frac{2}{3}\right)^{\frac{2}{3}} \left(\frac{2}{3} i {\left(\lambda + x\right)}^{\frac{3}{2}}\right)^{\frac{1}{3}} K_{2} \sqrt{\lambda + x} I_{\frac{1}{3}}(\frac{2}{3} {\left(\lambda + x\right)}^{\frac{3}{2}})}{2 {\left({\left(\lambda + x\right)}^{\frac{3}{2}}\right)}^{\frac{1}{3}}} + K_{1} \sqrt{\lambda + x} Y_{\frac{1}{3}}(\frac{2}{3} i {\left(\lambda + x\right)}^{\frac{3}{2}})$$
I am not qualified to check this solution.
