76.47.2 problem Ex. 2

Internal problem ID [20262]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 102
Problem number : Ex. 2
Date solved : Thursday, October 02, 2025 at 05:38:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{4} y^{\prime \prime }+x y^{\prime }+y&=\frac {1}{x} \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x^4*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 1/x; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 164

ode=x^4*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==1/x; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to e^{\frac {1}{2 x^2}} \left (420 x^6+45 x^4+6 x^2+1\right ) x^4 \left (1072 \sqrt {2 \pi } \text {erf}\left (\frac {1}{\sqrt {2} x}\right )+e^{-\frac {1}{2 x^2}} \left (1701 x^5-2142 x^3-\frac {1}{x^3}+2142 x-\frac {2}{x}\right )\right )+\frac {c_1 \left (1-x^2\right )}{x}+\frac {\left (1-x^2\right ) \left (70 x^6+\frac {45 x^4}{4}+3 x^2+\log (x)\right )}{x}+c_2 e^{\frac {1}{2 x^2}} \left (420 x^6+45 x^4+6 x^2+1\right ) x^4 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + y(x) - 1/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**4*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + y(x) - 1/x does not match hint 2nd_power_series_regular