34.4.1 problem 1

Internal problem ID [6055]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 1
Date solved : Monday, January 27, 2025 at 01:34:33 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+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(x*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \left (-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right )+c_2 \left (\frac {137 x^5}{432000}-\frac {25 x^4}{3456}+\frac {11 x^3}{108}-\frac {3 x^2}{4}+\left (-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)+2 x\right ) \]