56.4.52 problem 49

Internal problem ID [8941]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 49
Date solved : Monday, January 27, 2025 at 05:20:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}+6 x \right ) y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 44

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

\[ y = c_{1} \left (1-\frac {1}{6} x +\frac {1}{42} x^{2}-\frac {1}{336} x^{3}+\frac {1}{3024} x^{4}-\frac {1}{30240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (2880-2880 x +1440 x^{2}-480 x^{3}+120 x^{4}-24 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 68

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

\[ y(x)\to c_2 \left (\frac {x^4}{3024}-\frac {x^3}{336}+\frac {x^2}{42}-\frac {x}{6}+1\right )+c_1 \left (\frac {1}{x^5}-\frac {1}{x^4}+\frac {1}{2 x^3}-\frac {1}{6 x^2}+\frac {1}{24 x}\right ) \]