Internal problem ID [4211]
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: 9.
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 }-\left (x^{2}+4 x \right ) y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 61
Order:=6; dsolve(x^2*diff(y(x),x$2)-(x^2+4*x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{1} x^{3}+c_{2} \left (\left (6 x^{3}+6 x^{4}+3 x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \ln \relax (x )+\left (12-6 x +6 x^{2}+11 x^{3}+5 x^{4}+x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )\right ) x \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 74
AsymptoticDSolveValue[x^2*y''[x]-(x^2+4*x)*y'[x]+4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{2} (x+1) x^4 \log (x)+\frac {1}{4} \left (x^4+3 x^3+2 x^2-2 x+4\right ) x\right )+c_2 \left (\frac {x^8}{24}+\frac {x^7}{6}+\frac {x^6}{2}+x^5+x^4\right ) \]