20.26.24 problem 18

Internal problem ID [4049]
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 : 18
Date solved : Monday, January 27, 2025 at 08:07:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 62

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

\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \left (\frac {31 x^4+176 x^3+144 x^2-576 x+576}{576 x}-\frac {1}{48} x \left (x^2+8 x+24\right ) \log (x)\right )+c_2 \left (\frac {x^5}{8640}+\frac {x^4}{360}+\frac {x^3}{24}+\frac {x^2}{3}+x\right ) \]