88.25.6 problem 2 (f)

Internal problem ID [24211]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 212
Problem number : 2 (f)
Date solved : Thursday, October 02, 2025 at 10:00:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 111

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

\[ y = c_1 \,x^{-\sqrt {2}} \left (1+\frac {\sqrt {2}}{-24+24 \sqrt {2}} x^{2}-\frac {1}{5760} \frac {\sqrt {2}\, \left (4+\sqrt {2}\right )}{\left (-2+\sqrt {2}\right ) \left (\sqrt {2}-1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\sqrt {2}} \left (1+\frac {\sqrt {2}}{24+24 \sqrt {2}} x^{2}-\frac {1}{5760} \frac {\sqrt {2}\, \left (-4+\sqrt {2}\right )}{\left (2+\sqrt {2}\right ) \left (1+\sqrt {2}\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 274

ode=x^2*D[y[x],{x,2}]+Sin[x]*D[y[x],{x,1}]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {\left (\frac {1}{60 \sqrt {2}}-\frac {2-\sqrt {2}}{18 \sqrt {2} \left (\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )-\sqrt {2}\right )}\right ) x^4}{2-\sqrt {2}+\left (3-\sqrt {2}\right ) \left (4-\sqrt {2}\right )}-\frac {x^2}{3 \sqrt {2} \left (\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )-\sqrt {2}\right )}+1\right ) x^{-\sqrt {2}}+c_1 \left (\frac {\left (\frac {2+\sqrt {2}}{18 \sqrt {2} \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right )}-\frac {1}{60 \sqrt {2}}\right ) x^4}{2+\sqrt {2}+\left (3+\sqrt {2}\right ) \left (4+\sqrt {2}\right )}+\frac {x^2}{3 \sqrt {2} \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right )}+1\right ) x^{\sqrt {2}} \]

✗ Sympy

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

ValueError : ODE x**2*Derivative(y(x), (x, 2)) - 2*y(x) + sin(x)*Derivative(y(x), x) does not match