46.2.15 problem 17

Internal problem ID [7318]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 17
Date solved : Monday, January 27, 2025 at 02:50:16 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x y^{\prime \prime }+y^{\prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 44

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

\[ y = c_{1} x^{{3}/{4}} \left (1-\frac {8}{7} x +\frac {32}{77} x^{2}-\frac {256}{3465} x^{3}+\frac {512}{65835} x^{4}-\frac {4096}{7571025} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-8 x +\frac {32}{5} x^{2}-\frac {256}{135} x^{3}+\frac {512}{1755} x^{4}-\frac {4096}{149175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 83

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

\[ y(x)\to c_2 \left (-\frac {4096 x^5}{149175}+\frac {512 x^4}{1755}-\frac {256 x^3}{135}+\frac {32 x^2}{5}-8 x+1\right )+c_1 x^{3/4} \left (-\frac {4096 x^5}{7571025}+\frac {512 x^4}{65835}-\frac {256 x^3}{3465}+\frac {32 x^2}{77}-\frac {8 x}{7}+1\right ) \]