54.4.10 problem 10

Internal problem ID [8594]
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 : 10
Date solved : Monday, January 27, 2025 at 04:17:02 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x y^{\prime \prime }+\left (-2 x^{2}+1\right ) y^{\prime }-4 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 36

Order:=8; 
dsolve(2*x*diff(y(x),x$2)+(1-2*x^2)*diff(y(x),x)-4*x*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2}{3} x^{2}+\frac {4}{21} x^{4}+\frac {8}{231} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 61

AsymptoticDSolveValue[2*x*D[y[x],{x,2}]+(1-2*x^2)*D[y[x],x]-4*x*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {8 x^6}{231}+\frac {4 x^4}{21}+\frac {2 x^2}{3}+1\right ) \]