Internal problem ID [4207]
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: 5.
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 (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: 0.021 (sec). Leaf size: 65
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(x+1)*diff(y(x),x)+(3*x-1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {4}{3} x +\frac {5}{6} x^{2}-\frac {1}{3} x^{3}+\frac {7}{72} x^{4}-\frac {1}{45} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (6 x^{2}-8 x^{3}+5 x^{4}-2 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2-4 x +5 x^{2}+2 x^{3}-4 x^{4}+\frac {7}{3} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 85
AsymptoticDSolveValue[x^2*y''[x]+x*(x+1)*y'[x]+(3*x-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {13 x^4-12 x^3-4 x^2+8 x+4}{4 x}-\frac {1}{2} x \left (5 x^2-8 x+6\right ) \log (x)\right )+c_2 \left (\frac {7 x^5}{72}-\frac {x^4}{3}+\frac {5 x^3}{6}-\frac {4 x^2}{3}+x\right ) \]