54.4.17 problem 17

Internal problem ID [8601]
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 : 17
Date solved : Monday, January 27, 2025 at 04:17:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 36

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

\[ y = c_{1} x^{{3}/{2}} \left (1+\frac {2}{7} x^{2}+\frac {4}{77} x^{4}+\frac {8}{1155} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 61

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

\[ y(x)\to c_2 \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_1 \left (\frac {8 x^6}{1155}+\frac {4 x^4}{77}+\frac {2 x^2}{7}+1\right ) x^{3/2} \]