Internal problem ID [4770]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page
300
Problem number: 7.3.8 (d).
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 }+x y^{\prime }-{\mathrm e}^{x} y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 54
Order:=6; dsolve(x*diff(y(x),x$2)+x*diff(y(x),x)-exp(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1+\frac {1}{6} x^{2}+\frac {1}{72} x^{3}+\frac {7}{480} x^{4}+\frac {29}{10800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (x +\frac {1}{6} x^{3}+\frac {1}{72} x^{4}+\frac {7}{480} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-x -\frac {2}{9} x^{3}-\frac {11}{864} x^{4}-\frac {109}{4800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 70
AsymptoticDSolveValue[x*y''[x]+x*y'[x]-Exp[x]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {7 x^5}{480}+\frac {x^4}{72}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {1}{864} \left (-23 x^4-336 x^3-1728 x+864\right )+\frac {1}{72} x \left (x^3+12 x^2+72\right ) \log (x)\right ) \]