54.9.4 problem 4

Internal problem ID [8674]
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 : 4
Date solved : Monday, January 27, 2025 at 04:18:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }-x \left (2 x +7\right ) y^{\prime }+2 \left (5+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 55

Order:=8; 
dsolve(2*x^2*diff(y(x),x$2)-x*(2*x+7)*diff(y(x),x)+2*(x+5)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{2} \left (1+2 x +\frac {4}{3} x^{2}+\frac {8}{15} x^{3}+\frac {16}{105} x^{4}+\frac {32}{945} x^{5}+\frac {64}{10395} x^{6}+\frac {128}{135135} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{{5}/{2}} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 110

AsymptoticDSolveValue[2*x^2*D[y[x],{x,2}]-x*(2*x+7)*D[y[x],x]+2*(x+5)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {128 x^7}{135135}+\frac {64 x^6}{10395}+\frac {32 x^5}{945}+\frac {16 x^4}{105}+\frac {8 x^3}{15}+\frac {4 x^2}{3}+2 x+1\right ) x^2+c_1 \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) x^{5/2} \]