Internal problem ID [4208]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (-x^{2}+x \right ) y^{\prime \prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 58
Order:=6; dsolve((x-x^2)*diff(y(x),x$2)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1+\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {7}{48} x^{3}+\frac {91}{960} x^{4}+\frac {637}{9600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (x +\frac {1}{2} x^{2}+\frac {1}{4} x^{3}+\frac {7}{48} x^{4}+\frac {91}{960} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{4} x^{2}-\frac {1}{12} x^{3}-\frac {17}{576} x^{4}-\frac {311}{28800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 87
AsymptoticDSolveValue[(x-x^2)*y''[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{48} x \left (7 x^3+12 x^2+24 x+48\right ) \log (x)+\frac {1}{576} \left (-185 x^4-336 x^3-720 x^2-1152 x+576\right )\right )+c_2 \left (\frac {91 x^5}{960}+\frac {7 x^4}{48}+\frac {x^3}{4}+\frac {x^2}{2}+x\right ) \]