20.26.15 problem 7

Internal problem ID [4040]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 7
Date solved : Sunday, March 30, 2025 at 02:15:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 62

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

\[ y = \frac {c_1 \,x^{3} \left (1+\frac {1}{4} x -\frac {7}{40} x^{2}-\frac {37}{720} x^{3}+\frac {467}{20160} x^{4}+\frac {5647}{806400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x^{3}-\frac {1}{4} x^{4}+\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-6 x -3 x^{2}+3 x^{3}+\frac {29}{16} x^{4}-\frac {353}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 82

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

\[ y(x)\to c_1 \left (\frac {91 x^4+160 x^3-144 x^2-288 x+576}{576 x}-\frac {1}{48} x^2 (x+4) \log (x)\right )+c_2 \left (\frac {467 x^6}{20160}-\frac {37 x^5}{720}-\frac {7 x^4}{40}+\frac {x^3}{4}+x^2\right ) \]

✓ Sympy. Time used: 0.837 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {37 x^{3}}{720} - \frac {7 x^{2}}{40} + \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]