Internal problem ID [11757]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 20, Series solutions of second order linear equations. Exercises page
195
Problem number: 20.3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x \left (1-x \right ) y^{\prime \prime }-3 x y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 60
Order:=6; dsolve(x*(1-x)*diff(y(x),x$2)-3*x*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{2} +\left (1+3 x +5 x^{2}+7 x^{3}+9 x^{4}+11 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 63
AsymptoticDSolveValue[x*(1-x)*y''[x]-3*x*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (x^4+x^3+x^2+\left (4 x^3+3 x^2+2 x+1\right ) x \log (x)+x+1\right )+c_2 \left (5 x^5+4 x^4+3 x^3+2 x^2+x\right ) \]