54.8.3 problem 3

Internal problem ID [8665]
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 : 3
Date solved : Sunday, March 30, 2025 at 01:22:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 48

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}-\frac {1}{9} x^{3}+\frac {1}{64} x^{4}+\frac {13}{900} x^{5}+\frac {55}{20736} x^{6}-\frac {433}{705600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {2}{27} x^{3}-\frac {3}{128} x^{4}-\frac {253}{13500} x^{5}-\frac {95}{41472} x^{6}+\frac {153527}{148176000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 144

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

\[ y(x)\to c_1 \left (-\frac {433 x^7}{705600}+\frac {55 x^6}{20736}+\frac {13 x^5}{900}+\frac {x^4}{64}-\frac {x^3}{9}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {153527 x^7}{148176000}-\frac {95 x^6}{41472}-\frac {253 x^5}{13500}-\frac {3 x^4}{128}+\frac {2 x^3}{27}+\frac {x^2}{4}+\left (-\frac {433 x^7}{705600}+\frac {55 x^6}{20736}+\frac {13 x^5}{900}+\frac {x^4}{64}-\frac {x^3}{9}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.752 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {433 x^{7}}{705600} + \frac {55 x^{6}}{20736} + \frac {13 x^{5}}{900} + \frac {x^{4}}{64} - \frac {x^{3}}{9} - \frac {x^{2}}{4} + 1\right ) + O\left (x^{8}\right ) \]