Internal problem ID [1311]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 20.
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 \left (x +1\right ) y^{\prime }-\left (3 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 39
Order:=6; dsolve(3*x^2*diff(y(x),x$2)+x*(1+x)*diff(y(x),x)-(1+3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{3}} \left (1+\frac {2}{7} x +\frac {1}{70} x^{2}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {10}{3} x -\frac {35}{18} x^{2}-\frac {14}{81} x^{3}-\frac {7}{3888} x^{4}+\frac {7}{320760} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 65
AsymptoticDSolveValue[3*x^2*y''[x]+x*(1+x)*y'[x]-(1+3*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {x^2}{70}+\frac {2 x}{7}+1\right )+\frac {c_2 \left (\frac {7 x^5}{320760}-\frac {7 x^4}{3888}-\frac {14 x^3}{81}-\frac {35 x^2}{18}-\frac {10 x}{3}+1\right )}{\sqrt [3]{x}} \]