54.6.11 problem 11

Internal problem ID [8643]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 11
Date solved : Monday, January 27, 2025 at 04:18:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+4 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: 48

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

\[ y = c_{1} x^{5} \left (1+x +\frac {4}{7} x^{2}+\frac {5}{21} x^{3}+\frac {5}{63} x^{4}+\frac {1}{45} x^{5}+\frac {8}{1485} x^{6}+\frac {4}{3465} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (2880+2880 x +960 x^{2}+128 x^{5}+128 x^{6}+\frac {512}{7} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 76

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

\[ y(x)\to c_1 \left (\frac {2 x^6}{45}+\frac {2 x^5}{45}+\frac {x^2}{3}+x+1\right )+c_2 \left (\frac {8 x^{11}}{1485}+\frac {x^{10}}{45}+\frac {5 x^9}{63}+\frac {5 x^8}{21}+\frac {4 x^7}{7}+x^6+x^5\right ) \]