Internal problem ID [1295]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (5 x^{2}+3 x +3\right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1243
Order:=6; dsolve(2*x^2*(1+x+x^2)*diff(y(x),x$2)+x*(3+3*x+5*x^2)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-\frac {1}{4}-\frac {i \sqrt {7}}{4}} \left (1-\frac {1}{-2+i \sqrt {7}} x +\frac {1}{4} \frac {11+i \sqrt {7}}{\left (-2+i \sqrt {7}\right ) \left (i \sqrt {7}-4\right )} x^{2}+\frac {1}{12} \frac {-49 i \sqrt {7}+89}{\left (-2+i \sqrt {7}\right ) \left (i \sqrt {7}-4\right ) \left (i \sqrt {7}-6\right )} x^{3}+\frac {1}{48} \frac {-395 i \sqrt {7}-1553}{\left (-2+i \sqrt {7}\right ) \left (i \sqrt {7}-4\right ) \left (i \sqrt {7}-6\right ) \left (i \sqrt {7}-8\right )} x^{4}+\frac {1}{240} \frac {42423 i \sqrt {7}-45275}{\left (-2+i \sqrt {7}\right ) \left (i \sqrt {7}-4\right ) \left (i \sqrt {7}-6\right ) \left (i \sqrt {7}-8\right ) \left (i \sqrt {7}-10\right )} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{-\frac {1}{4}+\frac {i \sqrt {7}}{4}} \left (1+\frac {1}{2+i \sqrt {7}} x +\frac {1}{4} \frac {11-i \sqrt {7}}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right )} x^{2}+\frac {1}{12} \frac {-49 i \sqrt {7}-89}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right )} x^{3}+\frac {1}{48} \frac {395 i \sqrt {7}-1553}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right )} x^{4}+\frac {1}{240} \frac {42423 i \sqrt {7}+45275}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right ) \left (i \sqrt {7}+10\right )} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 4838
AsymptoticDSolveValue[2*x^2*(1+x+x^2)*y''[x]+x*(3+3*x+5*x^2)*y'[x]+y[x]==0,y[x],{x,0,5}]
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