73.25.13 problem 35.3 (g)

Internal problem ID [15721]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (g)
Date solved : Tuesday, January 28, 2025 at 08:05:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (4 x^{2}-1\right ) y^{\prime \prime }+\left (4-\frac {2}{x}\right ) y^{\prime }+\frac {\left (-x^{2}+1\right ) y}{x^{2}+1}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.077 (sec). Leaf size: 60

Order:=6; 
dsolve((4*x^2-1)*diff(y(x),x$2)+(4-2/x)*diff(y(x),x)+(1-x^2)/(1+x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+\frac {1}{6} x^{2}+\frac {1}{9} x^{3}+\frac {1}{24} x^{4}+\frac {31}{270} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +\ln \left (x \right ) \left (\left (-4\right ) x -\frac {2}{3} x^{3}-\frac {4}{9} x^{4}-\frac {1}{6} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1+4 x -\frac {7}{2} x^{2}+\frac {14}{9} x^{3}+\frac {133}{216} x^{4}+\frac {23}{90} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 79

AsymptoticDSolveValue[(4*x^2-1)*D[y[x],{x,2}]+(4-2/x)*D[y[x],x]+(1-x^2)/(1+x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{24}+\frac {x^3}{9}+\frac {x^2}{6}+1\right )+c_1 \left (\frac {229 x^4+480 x^3-756 x^2+1728 x+216}{216 x}-\frac {2}{9} \left (2 x^3+3 x^2+18\right ) \log (x)\right ) \]