78.19.2 problem 1 (b)

Internal problem ID [18421]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 1 (b)
Date solved : Tuesday, January 28, 2025 at 11:49:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 47

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

\[ y = c_{1} x^{1-i} \left (1+\left (-\frac {3}{5}-\frac {i}{5}\right ) x +\left (\frac {1}{5}-\frac {7 i}{20}\right ) x^{2}+\left (-\frac {337}{780}+\frac {161 i}{780}\right ) x^{3}+\left (\frac {1217}{6240}-\frac {1637 i}{6240}\right ) x^{4}+\left (-\frac {80549}{226200}+\frac {8367 i}{30160}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{1+i} \left (1+\left (-\frac {3}{5}+\frac {i}{5}\right ) x +\left (\frac {1}{5}+\frac {7 i}{20}\right ) x^{2}+\left (-\frac {337}{780}-\frac {161 i}{780}\right ) x^{3}+\left (\frac {1217}{6240}+\frac {1637 i}{6240}\right ) x^{4}+\left (-\frac {80549}{226200}-\frac {8367 i}{30160}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 94

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

\[ y(x)\to \left (\frac {1}{1248}+\frac {i}{6240}\right ) c_1 x^{1+i} \left ((297+268 i) x^4-(568+144 i) x^3+(324+372 i) x^2-(672-384 i) x+(1200-240 i)\right )-\left (\frac {1}{6240}+\frac {i}{1248}\right ) c_2 x^{1-i} \left ((268+297 i) x^4-(144+568 i) x^3+(372+324 i) x^2+(384-672 i) x-(240-1200 i)\right ) \]