78.19.4 problem 1 (d)

Internal problem ID [18423]
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 (d)
Date solved : Tuesday, January 28, 2025 at 11:49:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x +1\right ) x y^{\prime \prime }-\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: 0.009 (sec). Leaf size: 60

Order:=6; 
dsolve((3*x+1)*x*diff(y(x),x$2)-(x+1)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{2} \left (1-2 x +\frac {17}{4} x^{2}-\frac {289}{30} x^{3}+\frac {5491}{240} x^{4}-\frac {236113}{4200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (2 x^{2}-4 x^{3}+\frac {17}{2} x^{4}-\frac {289}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-4 x +6 x^{2}-12 x^{3}+\frac {209}{8} x^{4}-\frac {54247}{900} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \left (\frac {1}{16} \left (29 x^4-16 x^3+8 x^2+32 x+16\right )-\frac {1}{4} x^2 \left (17 x^2-8 x+4\right ) \log (x)\right )+c_2 \left (\frac {5491 x^6}{240}-\frac {289 x^5}{30}+\frac {17 x^4}{4}-2 x^3+x^2\right ) \]