54.9.11 problem 11

Internal problem ID [8681]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 11
Date solved : Monday, January 27, 2025 at 04:19:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-2 x \left (x +2\right ) y^{\prime }+\left (x +3\right ) 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: 54

Order:=8; 
dsolve(4*x^2*diff(y(x),x$2)-2*x*(2+x)*diff(y(x),x)+(3+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (1+\frac {1}{4} x +\frac {1}{24} x^{2}+\frac {1}{192} x^{3}+\frac {1}{1920} x^{4}+\frac {1}{23040} x^{5}+\frac {1}{322560} x^{6}+\frac {1}{5160960} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x c_{1} +\left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.103 (sec). Leaf size: 130

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]-2*x*(2+x)*D[y[x],x]+(3+x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^{13/2}}{46080}+\frac {x^{11/2}}{3840}+\frac {x^{9/2}}{384}+\frac {x^{7/2}}{48}+\frac {x^{5/2}}{8}+\frac {x^{3/2}}{2}+\sqrt {x}\right )+c_2 \left (\frac {x^{15/2}}{322560}+\frac {x^{13/2}}{23040}+\frac {x^{11/2}}{1920}+\frac {x^{9/2}}{192}+\frac {x^{7/2}}{24}+\frac {x^{5/2}}{4}+x^{3/2}\right ) \]