40.2.13 problem 15

Internal problem ID [8602]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 05:39:38 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x \left (1-x \right ) y^{\prime \prime }-\left (1+6 x \right ) y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {5}{2} x +\frac {35}{8} x^{2}+\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}+\frac {3003}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-2 x -8 x^{2}-16 x^{3}-\frac {128}{5} x^{4}-\frac {256}{7} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 79

ode=2*x*(1-x)*D[y[x],{x,2}]-(1+6*x)*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {256 x^5}{7}-\frac {128 x^4}{5}-16 x^3-8 x^2-2 x+1\right )+c_1 \left (\frac {3003 x^5}{256}+\frac {1155 x^4}{128}+\frac {105 x^3}{16}+\frac {35 x^2}{8}+\frac {5 x}{2}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.371 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} + C_{1} + O\left (x^{6}\right ) \]