6.13.3 problem 3

Internal problem ID [1894]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 05:20:59 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

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

\[ y = 1+x^{2}-\frac {2}{3} x^{3}+\frac {11}{6} x^{4}-\frac {9}{5} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

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

\[ y(x)\to -\frac {9 x^5}{5}+\frac {11 x^4}{6}-\frac {2 x^3}{3}+x^2+1 \]

✓ Sympy. Time used: 0.316 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{2} \left (\frac {11 x^{4}}{6} - \frac {2 x^{3}}{3} + x^{2} + 1\right ) + C_{1} x \left (- \frac {5 x^{3}}{2} + 2 x^{2} - x + 1\right ) + O\left (x^{6}\right ) \]