Internal problem ID [5734]
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(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+4 x^{2} y^{\prime }+2 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: 83
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+4*x^2*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{2}-\frac {i}{2}} \left (1-\frac {1}{2} x +\left (\frac {7}{40}+\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}-\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}+\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}-\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}+\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}-\frac {977 i}{48307200}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{2}+\frac {i}{2}} \left (1-\frac {1}{2} x +\left (\frac {7}{40}-\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}+\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}-\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}+\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}-\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}+\frac {977 i}{48307200}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 226
AsymptoticDSolveValue[4*x^2*y''[x]+4*x^2*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\left (\frac {3421}{14492160}-\frac {223 i}{1610240}\right ) x^{\frac {13}{2}+\frac {i}{2}}-\left (\frac {53}{32640}-\frac {13 i}{16320}\right ) x^{\frac {11}{2}+\frac {i}{2}}+\left (\frac {31}{3264}-\frac {i}{272}\right ) x^{\frac {9}{2}+\frac {i}{2}}-\left (\frac {11}{240}-\frac {i}{80}\right ) x^{\frac {7}{2}+\frac {i}{2}}+\left (\frac {7}{40}-\frac {i}{40}\right ) x^{\frac {5}{2}+\frac {i}{2}}-\frac {1}{2} x^{\frac {3}{2}+\frac {i}{2}}+x^{\frac {1}{2}+\frac {i}{2}}\right )+c_2 \left (\left (\frac {3421}{14492160}+\frac {223 i}{1610240}\right ) x^{\frac {13}{2}-\frac {i}{2}}-\left (\frac {53}{32640}+\frac {13 i}{16320}\right ) x^{\frac {11}{2}-\frac {i}{2}}+\left (\frac {31}{3264}+\frac {i}{272}\right ) x^{\frac {9}{2}-\frac {i}{2}}-\left (\frac {11}{240}+\frac {i}{80}\right ) x^{\frac {7}{2}-\frac {i}{2}}+\left (\frac {7}{40}+\frac {i}{40}\right ) x^{\frac {5}{2}-\frac {i}{2}}-\frac {1}{2} x^{\frac {3}{2}-\frac {i}{2}}+x^{\frac {1}{2}-\frac {i}{2}}\right ) \]