Internal problem ID [1406]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
II. Exercises 7.6. Page 374
Problem number: 59.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (4 x +3\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 49
Order:=6; dsolve(9*x^2*(3+x)*diff(y(x),x$2)+3*x*(3+7*x)*diff(y(x),x)+(3+4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{3} x +\frac {1}{9} x^{2}-\frac {1}{27} x^{3}+\frac {1}{81} x^{4}-\frac {1}{243} x^{5}\right ) x^{\frac {1}{3}} \left (\ln \relax (x ) c_{2}+c_{1}\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 92
AsymptoticDSolveValue[9*x^2*(3+x)*y''[x]+3*x*(3+7*x)*y'[x]+(3+4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^5}{243}+\frac {x^4}{81}-\frac {x^3}{27}+\frac {x^2}{9}-\frac {x}{3}+1\right )+c_2 \sqrt [3]{x} \left (-\frac {x^5}{243}+\frac {x^4}{81}-\frac {x^3}{27}+\frac {x^2}{9}-\frac {x}{3}+1\right ) \log (x) \]