54.4.18 problem 19

Internal problem ID [8602]
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 : 19
Date solved : Monday, January 27, 2025 at 04:17:13 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 20

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

\[ y = \frac {x^{{3}/{2}} c_{2} +c_{1}}{\sqrt {x}}+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 16

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

\[ y(x)\to c_1 x+\frac {c_2}{\sqrt {x}} \]