Internal problem ID [4726]
Book: A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd. New
York, reprinted 1956
Section: Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius. page
243
Problem number: Ex. 8(ii), page 258.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (x^{3}+1\right ) y^{\prime }+b x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 73
Order:=6; dsolve(x*diff(y(x),x$2)+(1+x*x^2)*diff(y(x),x)+b*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{4} b \,x^{2}+\frac {1}{64} b^{2} x^{4}+\frac {1}{50} b \,x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {b}{4} x^{2}-\frac {1}{9} x^{3}-\frac {3}{128} b^{2} x^{4}-\frac {61}{4500} b \,x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 103
AsymptoticDSolveValue[x*y''[x]+(1+x*x^2)*y'[x]+b*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {b^2 x^4}{64}+\frac {b x^5}{50}-\frac {b x^2}{4}+1\right )+c_2 \left (-\frac {3 b^2 x^4}{128}+\left (\frac {b^2 x^4}{64}+\frac {b x^5}{50}-\frac {b x^2}{4}+1\right ) \log (x)-\frac {61 b x^5}{4500}+\frac {b x^2}{4}-\frac {x^3}{9}\right ) \]