Internal problem ID [4924]
Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG,
EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 5. Series Solutions of ODEs. REVIEW QUESTIONS. page 201
Problem number: 19.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {y}{4 x}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 58
Order:=6; dsolve(diff(y(x),x$2)+1/(4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {1}{8} x +\frac {1}{192} x^{2}-\frac {1}{9216} x^{3}+\frac {1}{737280} x^{4}-\frac {1}{88473600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-\frac {1}{4} x +\frac {1}{32} x^{2}-\frac {1}{768} x^{3}+\frac {1}{36864} x^{4}-\frac {1}{2949120} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{64} x^{2}+\frac {7}{2304} x^{3}-\frac {35}{442368} x^{4}+\frac {101}{88473600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 85
AsymptoticDSolveValue[y''[x]+1/(4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x \left (x^3-48 x^2+1152 x-9216\right ) \log (x)}{36864}+\frac {-47 x^4+1920 x^3-34560 x^2+110592 x+442368}{442368}\right )+c_2 \left (\frac {x^5}{737280}-\frac {x^4}{9216}+\frac {x^3}{192}-\frac {x^2}{8}+x\right ) \]