36.6.12 problem 12

Internal problem ID [6373]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 12
Date solved : Sunday, March 30, 2025 at 10:53:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 76

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

\[ y = \left (1+\frac {\left (x +1\right )^{2}}{2}+\frac {2 \left (x +1\right )^{3}}{3}+\frac {11 \left (x +1\right )^{4}}{24}+\frac {\left (x +1\right )^{5}}{10}\right ) y \left (-1\right )+\left (x +1+2 \left (x +1\right )^{2}+\frac {7 \left (x +1\right )^{3}}{3}+\frac {3 \left (x +1\right )^{4}}{2}+\frac {4 \left (x +1\right )^{5}}{15}\right ) y^{\prime }\left (-1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{10} (x+1)^5+\frac {11}{24} (x+1)^4+\frac {2}{3} (x+1)^3+\frac {1}{2} (x+1)^2+1\right )+c_2 \left (\frac {4}{15} (x+1)^5+\frac {3}{2} (x+1)^4+\frac {7}{3} (x+1)^3+2 (x+1)^2+x+1\right ) \]

✓ Sympy. Time used: 0.712 (sec). Leaf size: 61

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

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