20.26.27 problem 21

Internal problem ID [4052]
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 : 21
Date solved : Monday, January 27, 2025 at 08:07:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\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: 62

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

\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (24 x^{3}+30 x^{4}+18 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \left (\frac {1}{2} x^2 (5 x+4) \log (x)-\frac {3 x^4-6 x^3-6 x^2+4 x-4}{4 x}\right )+c_2 \left (\frac {x^6}{12}+\frac {7 x^5}{24}+\frac {3 x^4}{4}+\frac {5 x^3}{4}+x^2\right ) \]