20.25.18 problem 19

Internal problem ID [4023]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 19
Date solved : Monday, January 27, 2025 at 08:06:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+x \left (3 x^{2}+1\right ) y^{\prime }-2 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 44

Order:=6; 
dsolve(3*x^2*diff(y(x),x$2)+x*(1+3*x^2)*diff(y(x),x)-2*x*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{{2}/{3}} \left (1+\frac {2}{5} x -\frac {3}{40} x^{2}-\frac {43}{660} x^{3}+\frac {31}{3696} x^{4}+\frac {2259}{261800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {1}{2} x^{2}-\frac {5}{21} x^{3}-\frac {73}{840} x^{4}+\frac {827}{27300} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 83

AsymptoticDSolveValue[3*x^2*D[y[x],{x,2}]+x*(1+3*x^2)*D[y[x],x]-2*x*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {827 x^5}{27300}-\frac {73 x^4}{840}-\frac {5 x^3}{21}+\frac {x^2}{2}+2 x+1\right )+c_1 x^{2/3} \left (\frac {2259 x^5}{261800}+\frac {31 x^4}{3696}-\frac {43 x^3}{660}-\frac {3 x^2}{40}+\frac {2 x}{5}+1\right ) \]