15.24.15 problem 15

Internal problem ID [3387]
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 : 15
Date solved : Monday, January 27, 2025 at 07:35:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (-x^{2}+3\right ) y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 46

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

\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1+\frac {1}{12} x^{2}+\frac {1}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (27 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-108 x^{2}-36 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 53

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

\[ y(x)\to c_2 \left (\frac {x^5}{128}+\frac {x^3}{12}+x\right )+c_1 \left (\frac {19 x^4+48 x^2+64}{64 x^3}-\frac {3}{16} x \log (x)\right ) \]