14.25.14 problem 15

Internal problem ID [4019]
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 11.4. page 758
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 07:00:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+2 \left (4 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 41

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)-x*(2*x+1)*diff(y(x),x)+2*(4*x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+3 x +\frac {21}{2} x^{2}-\frac {35}{2} x^{3}+\frac {35}{8} x^{4}-\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 \,x^{2} \left (1-\frac {4}{7} x +\frac {4}{63} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 65

ode=2*x^2*D[y[x],{x,2}]-x*(1+2*x)*D[y[x],x]+2*(4*x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {4 x^2}{63}-\frac {4 x}{7}+1\right ) x^2+\frac {c_2 \left (-\frac {7 x^5}{40}+\frac {35 x^4}{8}-\frac {35 x^3}{2}+\frac {21 x^2}{2}+3 x+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.378 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - x*(2*x + 1)*Derivative(y(x), x) + (8*x - 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {4 x^{2}}{63} - \frac {4 x}{7} + 1\right ) + \frac {C_{1} \left (- \frac {7 x^{5}}{40} + \frac {35 x^{4}}{8} - \frac {35 x^{3}}{2} + \frac {21 x^{2}}{2} + 3 x + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]