12.14.18 problem 18

Internal problem ID [1959]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 18
Date solved : Tuesday, March 04, 2025 at 01:46:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 45

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

\[ y = c_1 \sqrt {x}\, \left (1+2 x +\frac {2}{3} x^{2}+\frac {4}{45} x^{3}+\frac {2}{315} x^{4}+\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1+\frac {2}{3} x +\frac {2}{15} x^{2}+\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}+\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 84

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

\[ y(x)\to c_1 x \left (\frac {4 x^5}{155925}+\frac {2 x^4}{2835}+\frac {4 x^3}{315}+\frac {2 x^2}{15}+\frac {2 x}{3}+1\right )+c_2 \sqrt {x} \left (\frac {4 x^5}{14175}+\frac {2 x^4}{315}+\frac {4 x^3}{45}+\frac {2 x^2}{3}+2 x+1\right ) \]

✓ Sympy. Time used: 0.890 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {2 x^{4}}{2835} + \frac {4 x^{3}}{315} + \frac {2 x^{2}}{15} + \frac {2 x}{3} + 1\right ) + C_{1} \sqrt {x} \left (\frac {2 x^{4}}{315} + \frac {4 x^{3}}{45} + \frac {2 x^{2}}{3} + 2 x + 1\right ) + O\left (x^{6}\right ) \]