Internal problem ID [5708]
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 4.5. More on Regular
Singular Points. Page 183
Problem number: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-3 x y^{\prime }+\left (4+4 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 81
Order:=8; dsolve(x^2*diff(y(x),x$2)-3*x*diff(y(x),x)+(4*x+4)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-4 x +4 x^{2}-\frac {16}{9} x^{3}+\frac {4}{9} x^{4}-\frac {16}{225} x^{5}+\frac {16}{2025} x^{6}-\frac {64}{99225} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (8 x -12 x^{2}+\frac {176}{27} x^{3}-\frac {50}{27} x^{4}+\frac {1096}{3375} x^{5}-\frac {392}{10125} x^{6}+\frac {3872}{1157625} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 158
AsymptoticDSolveValue[x^2*y''[x]-3*x*y'[x]+(4*x+4)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {64 x^7}{99225}+\frac {16 x^6}{2025}-\frac {16 x^5}{225}+\frac {4 x^4}{9}-\frac {16 x^3}{9}+4 x^2-4 x+1\right ) x^2+c_2 \left (\left (\frac {3872 x^7}{1157625}-\frac {392 x^6}{10125}+\frac {1096 x^5}{3375}-\frac {50 x^4}{27}+\frac {176 x^3}{27}-12 x^2+8 x\right ) x^2+\left (-\frac {64 x^7}{99225}+\frac {16 x^6}{2025}-\frac {16 x^5}{225}+\frac {4 x^4}{9}-\frac {16 x^3}{9}+4 x^2-4 x+1\right ) x^2 \log (x)\right ) \]