Internal problem ID [2413]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page
758
Problem number: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+y^{\prime } {\mathrm e}^{x} x -y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+x*exp(x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {5}{4}} \left (1-\frac {1}{9} x -\frac {5}{468} x^{2}-\frac {11}{23868} x^{3}+\frac {79}{501228} x^{4}+\frac {16043}{313267500} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{4} x +\frac {5}{96} x^{2}+\frac {17}{8064} x^{3}-\frac {313}{1419264} x^{4}-\frac {69703}{709632000} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 86
AsymptoticDSolveValue[4*x^2*y''[x]+x*Exp[x]*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {16043 x^5}{313267500}+\frac {79 x^4}{501228}-\frac {11 x^3}{23868}-\frac {5 x^2}{468}-\frac {x}{9}+1\right )+\frac {c_2 \left (-\frac {69703 x^5}{709632000}-\frac {313 x^4}{1419264}+\frac {17 x^3}{8064}+\frac {5 x^2}{96}-\frac {x}{4}+1\right )}{\sqrt [4]{x}} \]