Internal problem ID [4497]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {\left (x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }-6 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 60
Order:=6; dsolve((x^2+x)*diff(y(x),x$2)+3*diff(y(x),x)-6*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1+\frac {3}{4} x^{2}-\frac {1}{10} x^{3}+\frac {17}{80} x^{4}-\frac {9}{100} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \relax (x ) \left (6 x^{2}+\frac {9}{2} x^{4}-\frac {3}{5} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2-12 x -24 x^{2}-22 x^{3}-\frac {171}{8} x^{4}-\frac {653}{100} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 73
AsymptoticDSolveValue[(x^2+x)*y''[x]+2*y'[x]-6*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {7 x^4}{20}-\frac {x^3}{6}+x^2+1\right )+c_1 \left (\frac {1}{3} \left (x^3-6 x^2-6\right ) \log (x)+\frac {7 x^4+240 x^3+72 x^2+180 x+36}{36 x}\right ) \]