Internal problem ID [5731]
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. Problems for review and discovert.
(A) Drill Exercises . Page 194
Problem number: 2(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) x^{2} y^{\prime \prime }-x y^{\prime }+\left (x +2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 87
Order:=8; dsolve((x^2+1)*x^2*diff(y(x),x$2)-x*diff(y(x),x)+(2+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{1-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}+\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}-\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}+\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}+\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}-\frac {106907 i}{17808000}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{1+i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}-\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}+\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}-\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}-\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}+\frac {106907 i}{17808000}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 122
AsymptoticDSolveValue[(x^2+1)*x^2*y''[x]-x*y'[x]+(2+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \left (\frac {1}{156000}+\frac {i}{1248000}\right ) c_2 x^{1-i} \left ((6080+10093 i) x^6-(10476-1572 i) x^5-(8220+19260 i) x^4+(21600+2400 i) x^3+(2400+50400 i) x^2-(38400+57600 i) x+(153600-19200 i)\right )-\left (\frac {1}{1248000}+\frac {i}{156000}\right ) c_1 x^{1+i} \left ((10093+6080 i) x^6+(1572-10476 i) x^5-(19260+8220 i) x^4+(2400+21600 i) x^3+(50400+2400 i) x^2-(57600+38400 i) x-(19200-153600 i)\right ) \]