56.4.38 problem 35

Internal problem ID [8927]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 35
Date solved : Monday, January 27, 2025 at 05:20:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 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: 44

Order:=6; 
dsolve(x*diff(y(x), x$2) +(1+x)*diff(y(x),x)+2*y(x) = 0,y(x),type='series',x=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {3}{2} x^{2}-\frac {2}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (3 x -\frac {13}{4} x^{2}+\frac {31}{18} x^{3}-\frac {173}{288} x^{4}+\frac {187}{1200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \left (-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right )+c_2 \left (\frac {187 x^5}{1200}-\frac {173 x^4}{288}+\frac {31 x^3}{18}-\frac {13 x^2}{4}+\left (-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right ) \log (x)+3 x\right ) \]