74.17.14 problem 14

Internal problem ID [16545]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 09:10:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 46

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

\[ y = x \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+2 x +2 x^{2}+\frac {11}{9} x^{3}+\frac {35}{72} x^{4}+\frac {103}{900} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {64}{27} x^{3}-\frac {497}{432} x^{4}-\frac {9371}{27000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 110

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

\[ y(x)\to c_1 x \left (\frac {103 x^5}{900}+\frac {35 x^4}{72}+\frac {11 x^3}{9}+2 x^2+2 x+1\right )+c_2 \left (x \left (-\frac {9371 x^5}{27000}-\frac {497 x^4}{432}-\frac {64 x^3}{27}-3 x^2-2 x\right )+x \left (\frac {103 x^5}{900}+\frac {35 x^4}{72}+\frac {11 x^3}{9}+2 x^2+2 x+1\right ) \log (x)\right ) \]