12.12.12 problem 12

Internal problem ID [1866]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 12
Date solved : Monday, January 27, 2025 at 05:37:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 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 )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

Order:=6; 
dsolve([(1+2*x^2)*diff(y(x),x$2)-9*x*diff(y(x),x)-6*y(x)=0,y(0) = 1, D(y)(0) = -1],y(x),type='series',x=0);
 

\[ y = 1-x +3 x^{2}-\frac {5}{2} x^{3}+5 x^{4}-\frac {21}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 32

AsymptoticDSolveValue[{(1+2*x^2)*D[y[x],{x,2}]-9*x*D[y[x],x]-6*y[x]==0,{y[0]==1,Derivative[1][y][0] ==-1}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to -\frac {21 x^5}{8}+5 x^4-\frac {5 x^3}{2}+3 x^2-x+1 \]