14.26.11 problem 3

Internal problem ID [4036]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 07:00:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 267

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

\[ y = c_1 \,x^{-\sqrt {2}} \left (1-2 \frac {1}{-1+2 \sqrt {2}} x +\frac {-5 \sqrt {2}+14}{40-24 \sqrt {2}} x^{2}+\frac {-122+75 \sqrt {2}}{684 \sqrt {2}-972} x^{3}+\frac {-1626 \sqrt {2}+2375}{52992-37440 \sqrt {2}} x^{4}+\frac {1}{7200} \frac {-75763+52810 \sqrt {2}}{\left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (-2+\sqrt {2}\right ) \left (-5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\sqrt {2}} \left (1+2 \frac {1}{1+2 \sqrt {2}} x +\frac {5 \sqrt {2}+14}{40+24 \sqrt {2}} x^{2}+\frac {122+75 \sqrt {2}}{684 \sqrt {2}+972} x^{3}+\frac {1626 \sqrt {2}+2375}{52992+37440 \sqrt {2}} x^{4}+\frac {1}{7200} \frac {75763+52810 \sqrt {2}}{\left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 2210

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

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✓ Sympy. Time used: 0.556 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{2} x^{\sqrt {2}} + \frac {C_{1}}{x^{\sqrt {2}}} + O\left (x^{6}\right ) \]