12.13.3 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 : Monday, January 27, 2025 at 05:37:58 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*}

Solution by Maple

Time used: 0.001 (sec). Leaf size: 18

Order:=6; 
dsolve([(1-2*x^2)*diff(y(x),x$2)+(2-6*x)*diff(y(x),x)-2*y(x)=0,y(0) = 1, D(y)(0) = 0],y(x),type='series',x=0);
 
\[ y = 1+x^{2}-\frac {2}{3} x^{3}+\frac {11}{6} x^{4}-\frac {9}{5} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Solution by Mathematica

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

AsymptoticDSolveValue[{(1-2*x^2)*D[y[x],{x,2}]+(2-6*x)*D[y[x],x]-2*y[x]==0,{y[0]==1,Derivative[1][y][0] ==0}},y[x],{x,0,"6"-1}]
 
\[ y(x)\to -\frac {9 x^5}{5}+\frac {11 x^4}{6}-\frac {2 x^3}{3}+x^2+1 \]