Internal problem ID [4718]
Internal file name [OUTPUT/4211_Sunday_June_05_2022_12_41_52_PM_98307444/index.tex]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 8.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_bessel_ode", "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_Emden, _Fowler]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {a y}{x^{\frac {3}{2}}}=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ y^{\prime \prime }+\frac {a y}{x^{\frac {3}{2}}} = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= 0\\ q(x) &= \frac {a}{x^{\frac {3}{2}}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([]\)
Irregular singular points : \([0, \infty ]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✗ Solution by Maple
Order:=6; dsolve(diff(y(x),x$2)+a/x^(3/2)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 576
AsymptoticDSolveValue[y''[x]+a/x^(3/2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to -\frac {16 x^5 \left (126 a^{10} c_2 \log (x)-252 \pi a^{10} c_1+504 \gamma a^{10} c_2-1423 a^{10} c_2+252 a^{10} c_2 \log (a)+504 a^{10} c_2 \log (2)\right )}{281302875 \pi }+\frac {32 x^{9/2} \left (1260 a^9 c_2 \log (x)-2520 \pi a^9 c_1+5040 \gamma a^9 c_2-13663 a^9 c_2+2520 a^9 c_2 \log (a)+5040 a^9 c_2 \log (2)\right )}{281302875 \pi }-\frac {8 x^4 \left (140 a^8 c_2 \log (x)-280 \pi a^8 c_1+560 \gamma a^8 c_2-1447 a^8 c_2+280 a^8 c_2 \log (a)+560 a^8 c_2 \log (2)\right )}{496125 \pi }+\frac {128 x^{7/2} \left (105 a^7 c_2 \log (x)-210 \pi a^7 c_1+420 \gamma a^7 c_2-1024 a^7 c_2+210 a^7 c_2 \log (a)+420 a^7 c_2 \log (2)\right )}{496125 \pi }-\frac {32 x^3 \left (15 a^6 c_2 \log (x)-30 \pi a^6 c_1+60 \gamma a^6 c_2-136 a^6 c_2+30 a^6 c_2 \log (a)+60 a^6 c_2 \log (2)\right )}{2025 \pi }+\frac {32 x^{5/2} \left (30 a^5 c_2 \log (x)-60 \pi a^5 c_1+120 \gamma a^5 c_2-247 a^5 c_2+60 a^5 c_2 \log (a)+120 a^5 c_2 \log (2)\right )}{675 \pi }-\frac {8 x^2 \left (6 a^4 c_2 \log (x)-12 \pi a^4 c_1+24 \gamma a^4 c_2-43 a^4 c_2+12 a^4 c_2 \log (a)+24 a^4 c_2 \log (2)\right )}{9 \pi }+\frac {32 x^{3/2} \left (3 a^3 c_2 \log (x)-6 \pi a^3 c_1+12 \gamma a^3 c_2-17 a^3 c_2+6 a^3 c_2 \log (a)+12 a^3 c_2 \log (2)\right )}{9 \pi }-\frac {8 x \left (a^2 c_2 \log (x)-2 \pi a^2 c_1+4 \gamma a^2 c_2-3 a^2 c_2+2 a^2 c_2 \log (a)+4 a^2 c_2 \log (2)\right )}{\pi }+\frac {8 a c_2 \sqrt {x}}{\pi }+\frac {2 c_2}{\pi } \]