15.23.20 problem 24

Internal problem ID [3370]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 24
Date solved : Monday, January 27, 2025 at 07:35:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+2 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: 0.011 (sec). Leaf size: 33

Order:=6; 
dsolve(3*x^2*diff(y(x),x$2)+2*x*diff(y(x),x)+(x^2-2)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{2}/{3}}}+c_{2} x \left (1-\frac {1}{22} x^{2}+\frac {1}{1496} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 48

AsymptoticDSolveValue[3*x^2*D[y[x],{x,2}]+2*x*D[y[x],x]+(x^2-2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 x \left (\frac {x^4}{1496}-\frac {x^2}{22}+1\right )+\frac {c_2 \left (\frac {x^4}{56}-\frac {x^2}{2}+1\right )}{x^{2/3}} \]