Internal problem ID [5703]
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: 4(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 52
Order:=8; dsolve(2*x*diff(y(x),x$2)+(x+1)*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {7}{6} x +\frac {21}{40} x^{2}-\frac {11}{80} x^{3}+\frac {143}{5760} x^{4}-\frac {13}{3840} x^{5}+\frac {17}{46080} x^{6}-\frac {323}{9676800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1-3 x +2 x^{2}-\frac {2}{3} x^{3}+\frac {1}{7} x^{4}-\frac {1}{45} x^{5}+\frac {4}{1485} x^{6}-\frac {4}{15015} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 106
AsymptoticDSolveValue[2*x*x*y''[x]+(x+1)*y'[x]+3*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {1386072 x^7}{35}+\frac {20088 x^6}{5}-\frac {2511 x^5}{5}+81 x^4-18 x^3+6 x^2-3 x+1\right )+c_2 e^{\left .\frac {1}{2}\right /x} \left (\frac {257243688 x^7}{35}+\frac {2381886 x^6}{5}+\frac {176436 x^5}{5}+3042 x^4+312 x^3+39 x^2+6 x+1\right ) x^{3/2} \]