12.16.6 problem 2

Internal problem ID [2068]
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 : 2
Date solved : Monday, January 27, 2025 at 05:41:30 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 58

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

\[ y = c_1 x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}-\frac {1}{144} x^{3}+\frac {1}{2880} x^{4}-\frac {1}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{144} x^{4}-\frac {1}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {7}{36} x^{3}-\frac {35}{1728} x^{4}+\frac {101}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{144} x \left (x^3-12 x^2+72 x-144\right ) \log (x)+\frac {-47 x^4+480 x^3-2160 x^2+1728 x+1728}{1728}\right )+c_2 \left (\frac {x^5}{2880}-\frac {x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]