54.5.14 problem 14

Internal problem ID [8629]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:21:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 54

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {5}{12} x^{3}+\frac {35}{192} x^{4}-\frac {21}{320} x^{5}+\frac {77}{3840} x^{6}-\frac {143}{26880} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {19}{128} x^{4}+\frac {25}{384} x^{5}-\frac {317}{13824} x^{6}+\frac {469}{69120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 161

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

\[ y(x)\to \frac {c_1 \left (-\frac {143 x^7}{26880}+\frac {77 x^6}{3840}-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right )}{x}+c_2 \left (\frac {\frac {469 x^7}{69120}-\frac {317 x^6}{13824}+\frac {25 x^5}{384}-\frac {19 x^4}{128}+\frac {x^3}{4}-\frac {x^2}{4}}{x}+\frac {\left (-\frac {143 x^7}{26880}+\frac {77 x^6}{3840}-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 0.856 (sec). Leaf size: 60

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

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {143 x^{8}}{114688} - \frac {143 x^{7}}{26880} + \frac {77 x^{6}}{3840} - \frac {21 x^{5}}{320} + \frac {35 x^{4}}{192} - \frac {5 x^{3}}{12} + \frac {3 x^{2}}{4} - x + 1\right )}{x} + O\left (x^{8}\right ) \]