28.4.13 problem 13

Internal problem ID [7202]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 04:25:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}&=\sqrt {x} \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)-(3*x+2)*diff(y(x),x)+(2*x-1)/x*y(x) = x^(1/2); 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 237

ode=2*x^2*D[y[x],{x,2}]-(3*x+2)*D[y[x],x]+(2*x-1)/x*y[x]==x^(1/2); 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {1}{128} i e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^3 \left (-5509056 i \sqrt {\pi } x \text {erfi}\left (\frac {1}{\sqrt {x}}\right )+\frac {i e^{\frac {1}{x}} \left (15663375 x^7+20072325 x^6+10329540 x^5+4131816 x^4+2754544 x^3+5509088 x^2-64 x-64\right )}{\sqrt {x}}+5509056 \sqrt {\pi } x\right )+c_2 e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^4+\frac {\left (\frac {5 x}{2}+1\right ) \left (-\frac {15015 x^6}{64}+\frac {693 x^5}{20}-\frac {189 x^4}{32}+\frac {7 x^3}{6}-\frac {x^2}{4}\right )}{\sqrt {x}}+\frac {c_1 \left (\frac {5 x}{2}+1\right )}{\sqrt {x}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(x) + 2*x**2*Derivative(y(x), (x, 2)) - (3*x + 2)*Derivative(y(x), x) + (2*x - 1)*y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -sqrt(x) + 2*x**2*Derivative(y(x), (x, 2)) - (3*x + 2)*Derivative(y(x), x) + (2*x - 1)*y(x)/x does not match hint 2nd_power_series_regular