20.26.6 problem (c)

Internal problem ID [4031]
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 11.5. page 771
Problem number : (c)
Date solved : Monday, January 27, 2025 at 08:06:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 39

Order:=7; 
dsolve(x^2*diff(y(x),x$2)+(9*x-2*x^5)*diff(y(x),x)+(25+5*x^2+10*x^4)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{-4-3 i} \left (1+\left (-\frac {1}{8}-\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}-\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}+\frac {3943 i}{29952}\right ) x^{6}+\operatorname {O}\left (x^{7}\right )\right )+c_{2} x^{-4+3 i} \left (1+\left (-\frac {1}{8}+\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}+\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}-\frac {3943 i}{29952}\right ) x^{6}+\operatorname {O}\left (x^{7}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 70

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+(9*x-2*x^5)*D[y[x],x]+(25+5*x^2+10*x^4)*y[x]==0,y[x],{x,0,"7"-1}]
 

\[ y(x)\to \left (\frac {1}{832}+\frac {5 i}{832}\right ) c_1 x^{-4+3 i} \left ((86+53 i) x^4+(56+32 i) x^2+(32-160 i)\right )-\left (\frac {5}{832}+\frac {i}{832}\right ) c_2 x^{-4-3 i} \left ((53+86 i) x^4+(32+56 i) x^2-(160-32 i)\right ) \]