Internal problem ID [8686]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1106.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+a y^{\prime }+b \,x^{\mathit {a1}} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 77
dsolve(x*diff(diff(y(x),x),x)+a*diff(y(x),x)+b*x^a1*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {a}{2}+\frac {1}{2}} \BesselJ \left (\frac {a -1}{\mathit {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\mathit {a1}}{2}+\frac {1}{2}}}{\mathit {a1} +1}\right )+c_{2} x^{-\frac {a}{2}+\frac {1}{2}} \BesselY \left (\frac {a -1}{\mathit {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\mathit {a1}}{2}+\frac {1}{2}}}{\mathit {a1} +1}\right ) \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 165
DSolve[b*x^a1*y[x] + a*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\frac {1}{\text {a1}}+1\right )^{\frac {a-1}{\text {a1}+1}} \text {a1}^{\frac {a-1}{\text {a1}+1}} b^{\frac {1-a}{2 \text {a1}+2}} \left (x^{\text {a1}}\right )^{-\frac {a-1}{2 \text {a1}}} \left (c_2 \text {Gamma}\left (\frac {-a+\text {a1}+2}{\text {a1}+1}\right ) J_{\frac {1-a}{\text {a1}+1}}\left (\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {\text {a1}+1}{2 \text {a1}}}}{\text {a1}+1}\right )+c_1 \text {Gamma}\left (\frac {a+\text {a1}}{\text {a1}+1}\right ) J_{\frac {a-1}{\text {a1}+1}}\left (\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {\text {a1}+1}{2 \text {a1}}}}{\text {a1}+1}\right )\right ) \\ \end{align*}