56.4.48 problem 45

Internal problem ID [8937]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 45
Date solved : Monday, January 27, 2025 at 05:20:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 34

Order:=6; 
dsolve(x^2*diff(y(x), x, x) + 4*x*diff(y(x), x) + (x^2+2)*y(x) = 0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 40

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(x^2+2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^3}{120}-\frac {x}{6}+\frac {1}{x}\right )+c_1 \left (\frac {x^2}{24}+\frac {1}{x^2}-\frac {1}{2}\right ) \]