28.5.14 problem 9.14

Internal problem ID [4601]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.14
Date solved : Tuesday, March 04, 2025 at 06:56:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(1/2*x+x^2)*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {4}{3} x^{2}-\frac {8}{15} x^{3}+\frac {16}{105} x^{4}-\frac {32}{945} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 81

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )+c_2 \left (-\frac {32 x^5}{945}+\frac {16 x^4}{105}-\frac {8 x^3}{15}+\frac {4 x^2}{3}-2 x+1\right ) \]

✓ Sympy. Time used: 0.972 (sec). Leaf size: 65

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {32 x^{5}}{945} + \frac {16 x^{4}}{105} - \frac {8 x^{3}}{15} + \frac {4 x^{2}}{3} - 2 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right ) + O\left (x^{6}\right ) \]