50.22.5 problem 1(e)

Internal problem ID [8148]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number : 1(e)
Date solved : Sunday, March 30, 2025 at 12:46:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+4\right ) y^{\prime \prime }-y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 79

Order:=8; 
ode:=(x^2+4)*diff(diff(y(x),x),x)-diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{8} x^{2}-\frac {1}{96} x^{3}+\frac {11}{1536} x^{4}+\frac {13}{10240} x^{5}-\frac {533}{737280} x^{6}-\frac {3809}{20643840} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{8} x^{2}-\frac {1}{32} x^{3}-\frac {5}{512} x^{4}+\frac {23}{10240} x^{5}+\frac {283}{245760} x^{6}-\frac {1649}{6881280} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 98

ode=(4+x^2)*D[y[x],{x,2}]-D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {3809 x^7}{20643840}-\frac {533 x^6}{737280}+\frac {13 x^5}{10240}+\frac {11 x^4}{1536}-\frac {x^3}{96}-\frac {x^2}{8}+1\right )+c_2 \left (-\frac {1649 x^7}{6881280}+\frac {283 x^6}{245760}+\frac {23 x^5}{10240}-\frac {5 x^4}{512}-\frac {x^3}{32}+\frac {x^2}{8}+x\right ) \]

✓ Sympy. Time used: 1.000 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 4)*Derivative(y(x), (x, 2)) + y(x) - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {533 x^{6}}{737280} + \frac {13 x^{5}}{10240} + \frac {11 x^{4}}{1536} - \frac {x^{3}}{96} - \frac {x^{2}}{8} + 1\right ) + C_{1} x \left (\frac {283 x^{5}}{245760} + \frac {23 x^{4}}{10240} - \frac {5 x^{3}}{512} - \frac {x^{2}}{32} + \frac {x}{8} + 1\right ) + O\left (x^{8}\right ) \]