Internal problem ID [1354]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (7 x^{2}+6 x +3\right ) y^{\prime }+\left (-3 x^{2}+6 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 81
Order:=8; dsolve(x^2*(1+x+2*x^2)*diff(y(x),x$2)+x*(3+6*x+7*x^2)*diff(y(x),x)+(1+6*x-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-2 x +\frac {9}{2} x^{2}-\frac {20}{3} x^{3}+\frac {173}{24} x^{4}-\frac {93}{20} x^{5}-\frac {419}{720} x^{6}+\frac {6697}{1260} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (x -\frac {15}{4} x^{2}+\frac {133}{18} x^{3}-\frac {3077}{288} x^{4}+\frac {4217}{400} x^{5}-\frac {70949}{14400} x^{6}-\frac {125221}{29400} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 162
AsymptoticDSolveValue[x^2*(1+x+2*x^2)*y''[x]+x*(3+6*x+7*x^2)*y'[x]+(1+6*x-3*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (\frac {6697 x^7}{1260}-\frac {419 x^6}{720}-\frac {93 x^5}{20}+\frac {173 x^4}{24}-\frac {20 x^3}{3}+\frac {9 x^2}{2}-2 x+1\right )}{x}+c_2 \left (\frac {-\frac {125221 x^7}{29400}-\frac {70949 x^6}{14400}+\frac {4217 x^5}{400}-\frac {3077 x^4}{288}+\frac {133 x^3}{18}-\frac {15 x^2}{4}+x}{x}+\frac {\left (\frac {6697 x^7}{1260}-\frac {419 x^6}{720}-\frac {93 x^5}{20}+\frac {173 x^4}{24}-\frac {20 x^3}{3}+\frac {9 x^2}{2}-2 x+1\right ) \log (x)}{x}\right ) \]