Internal problem ID [4224]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {u^{\prime \prime }-\frac {a^{2} u}{x^{\frac {2}{3}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 43
dsolve(diff(u(x),x$2)-a^2*x^(-2/3)*u(x)=0,u(x), singsol=all)
\[ u \relax (x ) = c_{1} \sqrt {x}\, \BesselJ \left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{\frac {2}{3}}}{2}\right )+c_{2} \sqrt {x}\, \BesselY \left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{\frac {2}{3}}}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 79
DSolve[u''[x]-a^2*x^(-2/3)*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
\begin{align*} u(x)\to \frac {3^{3/4} a^{3/4} \sqrt {x} \left (16 c_1 \text {Gamma}\left (\frac {5}{4}\right ) \text {BesselI}\left (-\frac {3}{4},\frac {3}{2} a x^{2/3}\right )+3 (-1)^{3/4} c_2 \text {Gamma}\left (\frac {3}{4}\right ) \text {BesselI}\left (\frac {3}{4},\frac {3}{2} a x^{2/3}\right )\right )}{8 \sqrt {2}} \\ \end{align*}