54.9.22 problem 23

Internal problem ID [8692]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 23
Date solved : Sunday, March 30, 2025 at 01:23:43 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 56

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}-\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {11}{36} x^{3}-\frac {25}{288} x^{4}+\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}+\frac {121}{235200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 162

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

\[ y(x)\to \frac {c_1 \left (-\frac {x^7}{5040}+\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )}{x}+c_2 \left (\frac {\frac {121 x^7}{235200}-\frac {49 x^6}{14400}+\frac {137 x^5}{7200}-\frac {25 x^4}{288}+\frac {11 x^3}{36}-\frac {3 x^2}{4}+x}{x}+\frac {\left (-\frac {x^7}{5040}+\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 0.911 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {x^{8}}{40320} - \frac {x^{7}}{5040} + \frac {x^{6}}{720} - \frac {x^{5}}{120} + \frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right )}{x} + O\left (x^{8}\right ) \]