20.25.16 problem 17

Internal problem ID [4021]
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 : 17
Date solved : Monday, January 27, 2025 at 08:06:24 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 3 x^{2} y^{\prime \prime }+x \left (7+3 x \right ) y^{\prime }+\left (1+6 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 47

Order:=6; 
dsolve(3*x^2*diff(y(x),x$2)+x*(7+3*x)*diff(y(x),x)+(1+6*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} \left (1-3 x +\frac {9}{4} x^{2}-\frac {27}{28} x^{3}+\frac {81}{280} x^{4}-\frac {243}{3640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 84

AsymptoticDSolveValue[3*x^2*D[y[x],{x,2}]+x*(7+3*x)*D[y[x],x]+(1+6*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )}{\sqrt [3]{x}}+\frac {c_2 \left (-\frac {243 x^5}{3640}+\frac {81 x^4}{280}-\frac {27 x^3}{28}+\frac {9 x^2}{4}-3 x+1\right )}{x} \]