54.7.6 problem 6

Internal problem ID [8654]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 6
Date solved : Monday, January 27, 2025 at 04:18:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }+8 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: 76

Order:=8; 
dsolve(x*diff(y(x),x$2)+(3+2*x)*diff(y(x),x)+8*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {8}{3} x +\frac {10}{3} x^{2}-\frac {8}{3} x^{3}+\frac {14}{9} x^{4}-\frac {32}{45} x^{5}+\frac {4}{15} x^{6}-\frac {16}{189} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (24 x^{2}-64 x^{3}+80 x^{4}-64 x^{5}+\frac {112}{3} x^{6}-\frac {256}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-8 x +20 x^{2}+16 x^{3}-64 x^{4}+\frac {224}{3} x^{5}-\frac {484}{9} x^{6}+\frac {6368}{225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.094 (sec). Leaf size: 116

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(3+2*x)*D[y[x],x]+8*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {4 x^6}{15}-\frac {32 x^5}{45}+\frac {14 x^4}{9}-\frac {8 x^3}{3}+\frac {10 x^2}{3}-\frac {8 x}{3}+1\right )+c_1 \left (\frac {326 x^6-480 x^5+468 x^4-216 x^3-36 x^2+36 x+9}{9 x^2}-\frac {4}{3} \left (14 x^4-24 x^3+30 x^2-24 x+9\right ) \log (x)\right ) \]