78.3.21 problem 6.c.i

Internal problem ID [21017]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 6.c.i
Date solved : Thursday, October 02, 2025 at 07:01:27 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+y x&=0 \end{align*}

Using series method with expansion around

\begin{align*} \infty \end{align*}

✗ Maple

Order:=6; 
ode:=diff(diff(y(x),x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=infinity);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 128

ode=D[y[x],{x,2}]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,Infinity,5}]
 

\[ y(x)\to c_1 e^{-\frac {2}{3} i x^{3/2}} \left (\frac {5 i}{48 x^{7/4}}-\frac {385}{4608 x^{13/4}}-\frac {85085 i}{663552 x^{19/4}}+\frac {37182145}{127401984 x^{25/4}}+\frac {1}{\sqrt [4]{x}}\right )+c_2 e^{\frac {2}{3} i x^{3/2}} \left (-\frac {5 i}{48 x^{7/4}}-\frac {385}{4608 x^{13/4}}+\frac {85085 i}{663552 x^{19/4}}+\frac {37182145}{127401984 x^{25/4}}+\frac {1}{\sqrt [4]{x}}\right ) \]

✓ Sympy. Time used: 0.344 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = C_{3} \left (\frac {C_{2}^{2} \left (C_{2} - x\right )^{4}}{24} - \frac {C_{2} \left (C_{2} - x\right )^{2}}{2} + \frac {\left (C_{2} - x\right )^{3}}{6} + 1\right ) + C_{1} \left (\frac {C_{2} \left (C_{2} - x\right )^{3}}{6} - C_{2} + x - \frac {\left (C_{2} - x\right )^{4}}{12}\right ) + O\left (x^{6}\right ) \]