Internal problem ID [4525]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-y^{\prime } {\mathrm e}^{x}+y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 1] \end {align*}
With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 16
Order:=6; dsolve([(x^2+1)*diff(y(x),x$2)-exp(x)*diff(y(x),x)+y(x)=0,y(0) = 1, D(y)(0) = 1],y(x),type='series',x=0);
\[ y \relax (x ) = 1+x +\frac {1}{24} x^{4}+\frac {1}{60} x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 20
AsymptoticDSolveValue[{(x^2+1)*y''[x]-Exp[x]*y'[x]+y[x]==0,{y[0]==1,y'[0]==1}},y[x],{x,0,5}]
\[ y(x)\to \frac {x^5}{60}+\frac {x^4}{24}+x+1 \]