Internal problem ID [414]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Chapter 11 Power series methods. Section 11.1 Introduction and Review of power series. Page
615
Problem number: problem 23.
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 }+y^{\prime } x^{2}+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 907
Order:=6; dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {i \sqrt {3}-3}{8 i \sqrt {3}-16} x^{2}+\frac {-i \sqrt {3}+5}{48 i \sqrt {3}-96} x^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}-5\right ) \left (i \sqrt {3}-7\right )}{\left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-2\right )} x^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}-7\right ) \left (i \sqrt {3}-9\right )}{\left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-2\right )} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {i \sqrt {3}+3}{8 i \sqrt {3}+16} x^{2}+\frac {-i \sqrt {3}-5}{48 i \sqrt {3}+96} x^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 886
AsymptoticDSolveValue[x^2*y''[x]+x^2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right ) x^5}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right ) \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right ) \left (1+\left (4-(-1)^{2/3}\right ) \left (5-(-1)^{2/3}\right )\right )}-\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right ) x^4}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right ) \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right )}+\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) x^3}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right )}-\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) x^2}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+\frac {(-1)^{2/3} x}{1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )}+1\right ) c_1 x^{-(-1)^{2/3}}+\left (-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right ) x^5}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right ) \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right ) \left (1+\left (4+\sqrt [3]{-1}\right ) \left (5+\sqrt [3]{-1}\right )\right )}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right ) x^4}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right ) \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right )}-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) x^3}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right )}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) x^2}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {\sqrt [3]{-1} x}{1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )}+1\right ) c_2 x^{\sqrt [3]{-1}} \]