34.4.10 problem 10

Internal problem ID [6064]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 10
Date solved : Monday, January 27, 2025 at 01:34:44 PM
CAS classification : [[_elliptic, _class_II]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}-\frac {3}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {1}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (-\frac {3 x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {x^4}{128}+\frac {x^2}{4}+\left (-\frac {3 x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]