34.4.7 problem 7

Internal problem ID [6061]
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 : 7
Date solved : Monday, January 27, 2025 at 01:34:40 PM
CAS classification : [[_elliptic, _class_I]]

\begin{align*} x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-3 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)+(1-3*x^2)*diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} x^{2}+\frac {9}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

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

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

\[ y(x)\to c_1 \left (\frac {9 x^4}{64}+\frac {x^2}{4}+1\right )+c_2 \left (\frac {21 x^4}{128}+\frac {x^2}{4}+\left (\frac {9 x^4}{64}+\frac {x^2}{4}+1\right ) \log (x)\right ) \]