15.24.2 problem 2

Internal problem ID [3374]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 2
Date solved : Monday, January 27, 2025 at 07:35:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+2 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 32

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

\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x^{2}-\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {x^4}{16}-\frac {x^2}{2}+1\right )+c_2 \left (-\frac {3 x^4}{32}+\frac {x^2}{2}+\left (\frac {x^4}{16}-\frac {x^2}{2}+1\right ) \log (x)\right ) \]