15.24.7 problem 7

Internal problem ID [3379]
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 : 7
Date solved : Monday, January 27, 2025 at 07:35:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (x^{2}-1\right ) y^{\prime }+\left (-x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 30

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*(x^2-1)*diff(y(x),x)+(1-x^2)*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 33

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

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