28.4.8 problem 8

Internal problem ID [7197]
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
Date solved : Tuesday, September 30, 2025 at 04:25:20 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+\frac {a y}{x^{{3}/{2}}}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✗ Maple

Order:=6; 
ode:=diff(diff(y(x),x),x)+a/x^(3/2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.165 (sec). Leaf size: 576

ode=D[y[x],{x,2}]+a/x^(3/2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {16 x^5 \left (126 a^{10} c_2 \log (x)-252 \pi a^{10} c_1+504 \gamma a^{10} c_2-1423 a^{10} c_2+252 a^{10} c_2 \log (a)+504 a^{10} c_2 \log (2)\right )}{281302875 \pi }+\frac {32 x^{9/2} \left (1260 a^9 c_2 \log (x)-2520 \pi a^9 c_1+5040 \gamma a^9 c_2-13663 a^9 c_2+2520 a^9 c_2 \log (a)+5040 a^9 c_2 \log (2)\right )}{281302875 \pi }-\frac {8 x^4 \left (140 a^8 c_2 \log (x)-280 \pi a^8 c_1+560 \gamma a^8 c_2-1447 a^8 c_2+280 a^8 c_2 \log (a)+560 a^8 c_2 \log (2)\right )}{496125 \pi }+\frac {128 x^{7/2} \left (105 a^7 c_2 \log (x)-210 \pi a^7 c_1+420 \gamma a^7 c_2-1024 a^7 c_2+210 a^7 c_2 \log (a)+420 a^7 c_2 \log (2)\right )}{496125 \pi }-\frac {32 x^3 \left (15 a^6 c_2 \log (x)-30 \pi a^6 c_1+60 \gamma a^6 c_2-136 a^6 c_2+30 a^6 c_2 \log (a)+60 a^6 c_2 \log (2)\right )}{2025 \pi }+\frac {32 x^{5/2} \left (30 a^5 c_2 \log (x)-60 \pi a^5 c_1+120 \gamma a^5 c_2-247 a^5 c_2+60 a^5 c_2 \log (a)+120 a^5 c_2 \log (2)\right )}{675 \pi }-\frac {8 x^2 \left (6 a^4 c_2 \log (x)-12 \pi a^4 c_1+24 \gamma a^4 c_2-43 a^4 c_2+12 a^4 c_2 \log (a)+24 a^4 c_2 \log (2)\right )}{9 \pi }+\frac {32 x^{3/2} \left (3 a^3 c_2 \log (x)-6 \pi a^3 c_1+12 \gamma a^3 c_2-17 a^3 c_2+6 a^3 c_2 \log (a)+12 a^3 c_2 \log (2)\right )}{9 \pi }-\frac {8 x \left (a^2 c_2 \log (x)-2 \pi a^2 c_1+4 \gamma a^2 c_2-3 a^2 c_2+2 a^2 c_2 \log (a)+4 a^2 c_2 \log (2)\right )}{\pi }+\frac {8 a c_2 \sqrt {x}}{\pi }+\frac {2 c_2}{\pi } \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)/x**(3/2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None