54.8.3 problem 3

Internal problem ID [8665]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page 391
Problem number : 3
Date solved : Monday, January 27, 2025 at 04:18:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+x \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: 48

Order:=8; 
dsolve(x*diff(y(x),x$2)+diff(y(x),x)+x*(1+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x^{2}-\frac {1}{9} x^{3}+\frac {1}{64} x^{4}+\frac {13}{900} x^{5}+\frac {55}{20736} x^{6}-\frac {433}{705600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {2}{27} x^{3}-\frac {3}{128} x^{4}-\frac {253}{13500} x^{5}-\frac {95}{41472} x^{6}+\frac {153527}{148176000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 144

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

\[ y(x)\to c_1 \left (-\frac {433 x^7}{705600}+\frac {55 x^6}{20736}+\frac {13 x^5}{900}+\frac {x^4}{64}-\frac {x^3}{9}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {153527 x^7}{148176000}-\frac {95 x^6}{41472}-\frac {253 x^5}{13500}-\frac {3 x^4}{128}+\frac {2 x^3}{27}+\frac {x^2}{4}+\left (-\frac {433 x^7}{705600}+\frac {55 x^6}{20736}+\frac {13 x^5}{900}+\frac {x^4}{64}-\frac {x^3}{9}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]