48.7.14 problem 14

Internal problem ID [9948]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 06:45:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1-2 x \right ) y^{\prime }-\left (x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 74

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+x*(1-2*x)*diff(y(x),x)-(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1+x +\frac {5}{8} x^{2}+\frac {7}{24} x^{3}+\frac {7}{64} x^{4}+\frac {11}{320} x^{5}+\frac {143}{15360} x^{6}+\frac {143}{64512} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x^{2}-x^{3}-\frac {5}{8} x^{4}-\frac {7}{24} x^{5}-\frac {7}{64} x^{6}-\frac {11}{320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-2 x +\frac {2}{3} x^{3}+\frac {61}{96} x^{4}+\frac {59}{160} x^{5}+\frac {919}{5760} x^{6}+\frac {449}{8064} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 115

ode=x^2*D[y[x],{x,2}]+x*(1-2*x)*D[y[x],x]-(x+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {1}{384} x \left (21 x^4+56 x^3+120 x^2+192 x+192\right ) \log (x)-\frac {617 x^6+1482 x^5+2730 x^4+3360 x^3+1440 x^2-5760 x-5760}{5760 x}\right )+c_2 \left (\frac {143 x^7}{15360}+\frac {11 x^6}{320}+\frac {7 x^5}{64}+\frac {7 x^4}{24}+\frac {5 x^3}{8}+x^2+x\right ) \]

✓ Sympy. Time used: 0.288 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - 2*x)*Derivative(y(x), x) - (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {143 x^{6}}{15360} + \frac {11 x^{5}}{320} + \frac {7 x^{4}}{64} + \frac {7 x^{3}}{24} + \frac {5 x^{2}}{8} + x + 1\right ) + O\left (x^{8}\right ) \]