Internal problem ID [6245]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page
391
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (6 x^{2}-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: 0.078 (sec). Leaf size: 53
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(1+x)*diff(y(x),x)-(1-3*x+6*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {4}{3} x +\frac {19}{12} x^{2}-\frac {7}{6} x^{3}+\frac {53}{72} x^{4}-\frac {116}{315} x^{5}+\frac {3247}{20160} x^{6}-\frac {5501}{90720} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2-4 x +5 x^{2}-\frac {44}{3} x^{3}+\frac {155}{12} x^{4}-\frac {331}{30} x^{5}+\frac {2321}{360} x^{6}-\frac {212}{63} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.235 (sec). Leaf size: 92
AsymptoticDSolveValue[x^2*y''[x]+x*(1+x)*y'[x]-(1-3*x+6*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {2321 x^5}{720}+\frac {331 x^4}{60}-\frac {155 x^3}{24}+\frac {22 x^2}{3}-\frac {5 x}{2}+\frac {1}{x}+2\right )+c_2 \left (\frac {3247 x^7}{20160}-\frac {116 x^6}{315}+\frac {53 x^5}{72}-\frac {7 x^4}{6}+\frac {19 x^3}{12}-\frac {4 x^2}{3}+x\right ) \]