12.11.7 problem 17

Internal problem ID [1846]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 17
Date solved : Tuesday, March 04, 2025 at 01:44:58 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 76

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

\[ y = \left (1+\frac {3 \left (x -2\right )^{2}}{4}-\frac {3 \left (x -2\right )^{3}}{8}+\frac {9 \left (x -2\right )^{4}}{32}-\frac {27 \left (x -2\right )^{5}}{160}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{2}}{2}+\frac {\left (x -2\right )^{3}}{2}-\frac {5 \left (x -2\right )^{4}}{16}+\frac {31 \left (x -2\right )^{5}}{160}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (-\frac {27}{160} (x-2)^5+\frac {9}{32} (x-2)^4-\frac {3}{8} (x-2)^3+\frac {3}{4} (x-2)^2+1\right )+c_2 \left (\frac {31}{160} (x-2)^5-\frac {5}{16} (x-2)^4+\frac {1}{2} (x-2)^3-\frac {1}{2} (x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 0.867 (sec). Leaf size: 61

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {5 \left (x - 2\right )^{4}}{16} + \frac {\left (x - 2\right )^{3}}{2} - \frac {\left (x - 2\right )^{2}}{2} - 2\right ) + C_{1} \left (\frac {9 \left (x - 2\right )^{4}}{32} - \frac {3 \left (x - 2\right )^{3}}{8} + \frac {3 \left (x - 2\right )^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]