54.6.15 problem 15

Internal problem ID [8647]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 15
Date solved : Monday, January 27, 2025 at 04:18:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 32

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

\[ y = c_{1} x^{2} \left (1-\frac {1}{5} x^{3}+\frac {1}{40} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-2+\frac {2}{3} x^{3}-\frac {1}{9} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {x^6}{18}-\frac {x^3}{3}+1\right )+c_2 \left (\frac {x^8}{40}-\frac {x^5}{5}+x^2\right ) \]