50.19.4 problem 1(d)

Internal problem ID [8112]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 1(d)
Date solved : Monday, January 27, 2025 at 03:43:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+3 x \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: 0.020 (sec). Leaf size: 72

Order:=8; 
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);
 

\[ y = c_{1} x^{2} \left (1-2 x +\frac {17}{4} x^{2}-\frac {289}{30} x^{3}+\frac {5491}{240} x^{4}-\frac {236113}{4200} x^{5}+\frac {28569673}{201600} x^{6}-\frac {28569673}{78400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (2 x^{2}-4 x^{3}+\frac {17}{2} x^{4}-\frac {289}{15} x^{5}+\frac {5491}{120} x^{6}-\frac {236113}{2100} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-4 x +6 x^{2}-12 x^{3}+\frac {209}{8} x^{4}-\frac {54247}{900} x^{5}+\frac {521849}{3600} x^{6}-\frac {158526173}{441000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.403 (sec). Leaf size: 118

AsymptoticDSolveValue[(3*x+1)*x*D[y[x],{x,2}]-(x+1)*D[y[x],x]+2*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {27353 x^6-12886 x^5+6525 x^4-3600 x^3+1800 x^2+7200 x+3600}{3600}-\frac {1}{240} x^2 \left (5491 x^4-2312 x^3+1020 x^2-480 x+240\right ) \log (x)\right )+c_2 \left (\frac {28569673 x^8}{201600}-\frac {236113 x^7}{4200}+\frac {5491 x^6}{240}-\frac {289 x^5}{30}+\frac {17 x^4}{4}-2 x^3+x^2\right ) \]