Internal problem ID [5732]
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(b).
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 }+x y^{\prime }+\left (1+x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 85
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}-\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}+\frac {223 i}{40281696000}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}+\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}-\frac {223 i}{40281696000}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 118
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(1+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \left (\frac {7}{36192000}+\frac {19 i}{54288000}\right ) c_1 x^i \left (i x^6+(12-36 i) x^5-(660-780 i) x^4+(16800-7200 i) x^3-(194400+36000 i) x^2+(633600+921600 i) x+(1209600-2188800 i)\right )-\left (\frac {19}{54288000}+\frac {7 i}{36192000}\right ) c_2 x^{-i} \left (x^6-(36-12 i) x^5+(780-660 i) x^4-(7200-16800 i) x^3-(36000+194400 i) x^2+(921600+633600 i) x-(2188800-1209600 i)\right ) \]