14.25.6 problem 7

Internal problem ID [4011]
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 11.4. page 758
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 07:00:21 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 45

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+x*exp(x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{4} x +\frac {5}{96} x^{2}+\frac {17}{8064} x^{3}-\frac {313}{1419264} x^{4}-\frac {69703}{709632000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{4}}}+c_2 x \left (1-\frac {1}{9} x -\frac {5}{468} x^{2}-\frac {11}{23868} x^{3}+\frac {79}{501228} x^{4}+\frac {16043}{313267500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 86

ode=4*x^2*D[y[x],{x,2}]+x*Exp[x]*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (\frac {16043 x^5}{313267500}+\frac {79 x^4}{501228}-\frac {11 x^3}{23868}-\frac {5 x^2}{468}-\frac {x}{9}+1\right )+\frac {c_2 \left (-\frac {69703 x^5}{709632000}-\frac {313 x^4}{1419264}+\frac {17 x^3}{8064}+\frac {5 x^2}{96}-\frac {x}{4}+1\right )}{\sqrt [4]{x}} \]

✓ Sympy. Time used: 0.305 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + x*exp(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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