12.16.35 problem 31

Internal problem ID [2097]
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 : 31
Date solved : Monday, January 27, 2025 at 05:42:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\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: 36

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+8*x*diff(y(x),x)-(35-x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{6} \left (1-\frac {1}{64} x^{2}+\frac {1}{10240} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-86400-2700 x^{2}-\frac {675}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{7}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 58

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+8*x*D[y[x],x]-(35-x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{32 x^{3/2}}+\frac {1}{x^{7/2}}+\frac {\sqrt {x}}{1024}\right )+c_2 \left (\frac {x^{13/2}}{10240}-\frac {x^{9/2}}{64}+x^{5/2}\right ) \]