54.8.1 problem 1

Internal problem ID [8663]
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 : 1
Date solved : Monday, January 27, 2025 at 04:18:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (x^{3}+x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 56

Order:=8; 
dsolve(x^2*diff(y(x),x$2)+3*x*diff(y(x),x)+(1+x+x^3)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {5}{36} x^{3}+\frac {41}{576} x^{4}-\frac {37}{2880} x^{5}+\frac {437}{103680} x^{6}-\frac {7817}{5080320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {19}{108} x^{3}-\frac {593}{3456} x^{4}+\frac {3629}{86400} x^{5}-\frac {7733}{1036800} x^{6}+\frac {485257}{118540800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 164

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

\[ y(x)\to \frac {c_1 \left (-\frac {7817 x^7}{5080320}+\frac {437 x^6}{103680}-\frac {37 x^5}{2880}+\frac {41 x^4}{576}-\frac {5 x^3}{36}+\frac {x^2}{4}-x+1\right )}{x}+c_2 \left (\frac {\frac {485257 x^7}{118540800}-\frac {7733 x^6}{1036800}+\frac {3629 x^5}{86400}-\frac {593 x^4}{3456}+\frac {19 x^3}{108}-\frac {3 x^2}{4}+2 x}{x}+\frac {\left (-\frac {7817 x^7}{5080320}+\frac {437 x^6}{103680}-\frac {37 x^5}{2880}+\frac {41 x^4}{576}-\frac {5 x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)}{x}\right ) \]