67.25.9 problem 35.3 (c)

Internal problem ID [17004]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (c)
Date solved : Thursday, October 02, 2025 at 01:41:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{4}+x^{3}\right ) y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+827 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 62

Order:=6; 
ode:=(-x^4+x^3)*diff(diff(y(x),x),x)+(3*x-1)*diff(y(x),x)+827*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = c_1 \left (x -1\right )^{3} \left (1+\frac {409}{2} \left (x -1\right )+\frac {328391}{20} \left (x -1\right )^{2}+\frac {128327201}{180} \left (x -1\right )^{3}+\frac {19341852779}{1008} \left (x -1\right )^{4}+\frac {6949904889503}{20160} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+c_2 \left (\ln \left (x -1\right ) \left (567661070 \left (x -1\right )^{3}[phone] \left (x -1\right )^{4}+\frac {18641478643837}{2} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+\left (12-4962 \left (x -1\right )+2059230 \left (x -1\right )^{2}-6162812 \left (x -1\right )^{3}-\frac {592298912511}{4} \left (x -1\right )^{4}-\frac {744988601770307}{40} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 105

ode=(x^3-x^4)*D[y[x],{x,2}]+(3*x-1)*D[y[x],x]+827*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_2 \left (\frac {19341852779 (x-1)^7}{1008}+\frac {128327201}{180} (x-1)^6+\frac {328391}{20} (x-1)^5+\frac {409}{2} (x-1)^4+(x-1)^3\right )+c_1 \left (\frac {1}{144} \left (-2226119942329 (x-1)^4-2270644232 (x-1)^3+24710760 (x-1)^2-59544 (x-1)+144\right )+\frac {283830535}{12} (409 (x-1)+2) (x-1)^3 \log (x-1)\right ) \]

✓ Sympy. Time used: 0.415 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x - 1)*Derivative(y(x), x) + (-x**4 + x**3)*Derivative(y(x), (x, 2)) + 827*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - 1\right )^{3} + C_{1} + O\left (x^{6}\right ) \]