12.16.11 problem 7

Internal problem ID [2073]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 7
Date solved : Monday, January 27, 2025 at 05:41:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (9-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 60

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+4*x*diff(y(x),x)-(9-x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {1}{16} x +\frac {1}{640} x^{2}-\frac {1}{46080} x^{3}+\frac {1}{5160960} x^{4}-\frac {1}{825753600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {1}{64} x^{3}+\frac {1}{1024} x^{4}-\frac {1}{40960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {3}{16} x^{2}-\frac {5}{4096} x^{4}+\frac {39}{819200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 96

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+4*x*D[y[x],x]-(9-x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^{11/2}}{5160960}-\frac {x^{9/2}}{46080}+\frac {x^{7/2}}{640}-\frac {x^{5/2}}{16}+x^{3/2}\right )+c_1 \left (\frac {(x-16) x^{3/2} \log (x)}{12288}-\frac {19 x^4-64 x^3-2304 x^2-18432 x-147456}{147456 x^{3/2}}\right ) \]