Internal problem ID [6175]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {3 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: 0.02 (sec). Leaf size: 53
Order:=8; dsolve(3*x^2*diff(y(x),x$2)+x*diff(y(x),x)-(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{3}} \left (1+\frac {1}{7} x +\frac {1}{140} x^{2}+\frac {1}{5460} x^{3}+\frac {1}{349440} x^{4}+\frac {1}{33196800} x^{5}+\frac {1}{4381977600} x^{6}+\frac {1}{766846080000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{1} \left (1-x -\frac {1}{4} x^{2}-\frac {1}{60} x^{3}-\frac {1}{1920} x^{4}-\frac {1}{105600} x^{5}-\frac {1}{8870400} x^{6}-\frac {1}{1055577600} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 112
AsymptoticDSolveValue[3*x^2*y''[x]+x*y'[x]-(1+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (\frac {x^7}{766846080000}+\frac {x^6}{4381977600}+\frac {x^5}{33196800}+\frac {x^4}{349440}+\frac {x^3}{5460}+\frac {x^2}{140}+\frac {x}{7}+1\right )+\frac {c_2 \left (-\frac {x^7}{1055577600}-\frac {x^6}{8870400}-\frac {x^5}{105600}-\frac {x^4}{1920}-\frac {x^3}{60}-\frac {x^2}{4}-x+1\right )}{\sqrt [3]{x}} \]