Internal problem ID [1297]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+x \left (7 x^{2}+8 x +5\right ) y^{\prime }-\left (-9 x^{2}-2 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 43
Order:=6; dsolve(x^2*(3+3*x+x^2)*diff(y(x),x$2)+x*(5+8*x+7*x^2)*diff(y(x),x)-(1-2*x-9*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {4}{7} x -\frac {7}{45} x^{2}+\frac {970}{2457} x^{3}-\frac {5707}{22680} x^{4}+\frac {13568}{300105} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-x^{2}+\frac {2}{3} x^{3}-\frac {10}{33} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 74
AsymptoticDSolveValue[x^2*(3+3*x+x^2)*y''[x]+x*(5+8*x+7*x^2)*y'[x]-(1-2*x-9*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_2 \left (-\frac {10 x^5}{33}+\frac {2 x^3}{3}-x^2+1\right )}{x}+c_1 \sqrt [3]{x} \left (\frac {13568 x^5}{300105}-\frac {5707 x^4}{22680}+\frac {970 x^3}{2457}-\frac {7 x^2}{45}-\frac {4 x}{7}+1\right ) \]