Internal problem ID [14823]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
Order:=6; dsolve(x*(1-x)*diff(y(x),x$2)+diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-2 x +x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (3 x -3 x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 59
AsymptoticDSolveValue[x*(1-x)*y''[x]+y'[x]+2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (x^2-2 x+1\right )+c_2 \left (\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{3}-3 x^2+\left (x^2-2 x+1\right ) \log (x)+3 x\right ) \]