54.4.7 problem 7

Internal problem ID [8591]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 7
Date solved : Monday, January 27, 2025 at 04:16:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} y^{\prime \prime }+10 x y^{\prime }-y \left (1+x \right )&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 56

Order:=8; 
dsolve(8*x^2*diff(y(x),x$2)+10*x*diff(y(x),x)-(1+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{2} x^{{3}/{4}} \left (1+\frac {1}{14} x +\frac {1}{616} x^{2}+\frac {1}{55440} x^{3}+\frac {1}{8426880} x^{4}+\frac {1}{1938182400} x^{5}+\frac {1}{627971097600} x^{6}+\frac {1}{272539456358400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1+\frac {1}{2} x +\frac {1}{40} x^{2}+\frac {1}{2160} x^{3}+\frac {1}{224640} x^{4}+\frac {1}{38188800} x^{5}+\frac {1}{9623577600} x^{6}+\frac {1}{3368252160000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 118

AsymptoticDSolveValue[8*x^2*D[y[x],{x,2}]+10*x*D[y[x],x]-(1+x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {x^7}{272539456358400}+\frac {x^6}{627971097600}+\frac {x^5}{1938182400}+\frac {x^4}{8426880}+\frac {x^3}{55440}+\frac {x^2}{616}+\frac {x}{14}+1\right )+\frac {c_2 \left (\frac {x^7}{3368252160000}+\frac {x^6}{9623577600}+\frac {x^5}{38188800}+\frac {x^4}{224640}+\frac {x^3}{2160}+\frac {x^2}{40}+\frac {x}{2}+1\right )}{\sqrt {x}} \]