54.8.4 problem 4

Internal problem ID [8666]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page 391
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 06:11:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (6 x^{2}-3 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 53

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+x*(1+x)*diff(y(x),x)-(6*x^2-3*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x \left (1-\frac {4}{3} x +\frac {19}{12} x^{2}-\frac {7}{6} x^{3}+\frac {53}{72} x^{4}-\frac {116}{315} x^{5}+\frac {3247}{20160} x^{6}-\frac {5501}{90720} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2-4 x +5 x^{2}-\frac {44}{3} x^{3}+\frac {155}{12} x^{4}-\frac {331}{30} x^{5}+\frac {2321}{360} x^{6}-\frac {212}{63} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 92

ode=x^2*D[y[x],{x,2}]+x*(1+x)*D[y[x],x]-(1-3*x+6*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {2321 x^5}{720}+\frac {331 x^4}{60}-\frac {155 x^3}{24}+\frac {22 x^2}{3}-\frac {5 x}{2}+\frac {1}{x}+2\right )+c_2 \left (\frac {3247 x^7}{20160}-\frac {116 x^6}{315}+\frac {53 x^5}{72}-\frac {7 x^4}{6}+\frac {19 x^3}{12}-\frac {4 x^2}{3}+x\right ) \]

✓ Sympy. Time used: 1.106 (sec). Leaf size: 97

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(x + 1)*Derivative(y(x), x) - (6*x**2 - 3*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {3247 x^{6}}{20160} - \frac {116 x^{5}}{315} + \frac {53 x^{4}}{72} - \frac {7 x^{3}}{6} + \frac {19 x^{2}}{12} - \frac {4 x}{3} + 1\right ) + \frac {C_{1} \left (- \frac {1061 x^{8}}{3360} + \frac {16 x^{7}}{21} - \frac {83 x^{6}}{60} + \frac {13 x^{5}}{5} - \frac {5 x^{4}}{2} + 4 x^{3} + 2 x + 1\right )}{x} + O\left (x^{8}\right ) \]